TKEATISE 


ON  THE 


PRINCIPLES  AND  APPLICATIONS 


OF 


ANALYTIC  GEOMETRY. 


BY 

HENRY  T.  EDDY,  C.E.,  PH.D., 

PROFESSOR  OF  MATHEMATICS  AND  ASTRONOMY  IN  THE  UNIVERSITY 
OF  CINCINNATI. 


PHILADELPHIA: 

COWPERTHWAIT  &  COMPANY. 

1874. 


Entered  according  to  Act  of  Congress,  in  the  year  187 U,  by 

HENRY  T.  EDDY, 
/Z^73J 

In  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


WESTCOTT  &  THOMSON,  EDMUND  DEACON, 

Stereotype™  and  Electrotypers,  Phila.  Printer,  Phila. 


PREFACE. 


THE  following  treatise,  designed  as  a  text-book  upon  Analytic  Geometry,  has 
been  written  with  the  most  practical  ends  in  view,  and  is  intended  to  meet  the 
wants  of  classes  in  Scientific  and  Technological  Schools,  Colleges  and  Universi- 
ties. While  the  needs  of  the  student  of  Mechanics,  Astronomy  and  Civil 
Engineering  have  never  been  forgotten,  it  has  been  found  possible  to  so  select 
the  material  and  to  put  it  in  such  shape  as  to  adapt  the  work- to  the  student 
who  pursues  the  subject  merely  as  a  part  of  a  liberal  education. 

The  prime  difficulty  the  ordinary  student  meets  in  the  'study  of  analytic 
geometry  is  in  the  use  of  variables,  since  with  these  he  has  had  no  previous 
acquaintance. 

No  pains  has  been  spared  to  make  the  introduction  to  their  use  clear  and  free 
from  all  other  complexities.  To  this  end  a  thorough  knowledge  of  co-ordinates 
has  been  first  secured  by  the  study  of  the  relations  of  points,  the  transformation 
of  co-ordinates,  etc. 

Again,  the  entire  subject  of  the  general  relation  of  constant  and  variable 
quantities  is  postponed  to  Chapter  V,  at  which  point  the  student  will  have 
attained  a  sufficient  acquaintance  with  the  processes  and  notation  peculiar  to 
analytic  geometry  to  grasp  the  ideas  advanced  and  use  them  in  after  work. 

To  secure  an  accurate  knowledge  of  the  meaning  of  the  general  equations,  it 
is  essential  that  the  student  should  solve  numerous  numerical  examples.  They 
should  be  illustrations,  and  of  such  simple  character  as  to  be  readily  solved  by 
any  one  who  understands  the  preceding  text. 

Such  are  the  examples  interspersed  through  the  work,  which  should  in  no 
case  be  omitted.  Indeed,  if  the  class  is  numerous,  the  teacher  is  advised  to 
largely  increase  the  number  of  examples  as  class-room  work  by  substituting 
other  numbers  than  those  used,  and  giving  each  example  to  a  sufficient  number 
of  different  computers  to  ensure  correct  results. 

The  Exercises  are  much  more  difficult  than  the  examples,  and  have  two 
objects  in  view :  first,  as  original  work  for  the  more  ambitious  students ;  and 
secondly,  as  results  to  be  referred  to  in  the  students'  subsequent  studies.  They 
may  be  omitted  by  the  ordinary  student. 

The  great  difficulty  which  the  teacher  experiences  is  not  usually  that  the 


4  PREFACE. 

student  cannot  be  made  to  apprehend  the  true  import  of  the  demonstrations, 
but  it  is  this — that  he  afterward  fails  to  recall  the  necessary  equations  and  their 
significance. 

To  assist  the  teacher  in  this  vital  point,  the  statement  of  each  theorem  is  in  a 
form  to  be  memorized,  and  contains  some  important  equation  and  its  signifi- 
cation. The  importance  of  acquiring  a  perfect  familiarity  with  these  statements 
in  algebraic  language  instead  of  ordinary  language  cannot  be  too  strongly 
emphasized.  It  has  been  found  by  the  best  teachers  that  ten  or  fifteen  minutes 
during  every  recitation  hour  should  be  spent  in  reciting  from  memory  the  state- 
ments of  all  theorems  previously  studied. 

The  form  of  notation  adopted  is  thoroughly  systematized,  and  prepares  the 
student  to  read  with  ease  the  great  modern  writers  upon  analytic  geometry. 
The  marked  value  of  the  angular  notation  used  is  a  sufficient  recommendation 
for  its  adoption.  For  it  I  am  happy  to  acknowledge  my  indebtedness  to  Prof. 
J.  M.  Peirce,  of  Harvard  University,  from  whose  works  it  is  borrowed. 

The  one  great  defect  of  text-books  upon  analytic  geometry  is  the  omission  of 
general  principles.  It  appears  to  be  assumed  that  an  acquaintance  with  its 
ordinary  processes  gives  the  student  a  knowledge  of  its  principles.  This  is 
far  from  being  a  correct  assumption,  as  an  examination  of  the  general  princi- 
ples stated  and  demonstrated  in  Chapter  V  will  abundantly  show. 

The  general  discussion  of  curves  and  their  singularities  by  means  of  their 
approximate  equations — a  method  due  to  the  genius  of  Newton — is  here  for  the 
first  time  rendered  accessible  to  the  ordinary  student,  and  it  is  thought  that  it 
will  serve  a  most  useful  purpose  by  putting  into  his  hands  an  instrument  of 
research  of  practical  value  whose  power  compares  favorably  with  the  resources 
of  the  differential  calculus. 

No  attempt  has  been  made  in  the  last  chapters  to  follow  the  beaten  track  of 
previous  text-books,  but  rather  to  select  matter  respecting  spirals,  etc.,  of  the 
greatest  use  to  the  student. 

The  book  will  be  found  to  be  suited  to  the  wants  of  classes  taking  either  a 
longer  or  shorter  course  by  the  various  omissions  indicated  in  the  course  of  it. 

I  take  this  opportunity  to  express  my  thanks  to  Prof.  James  Edward  Oliver, 
of  Cornell  University,  for  many  happy  suggestions. 

I  am  especially  indebted  to  Prof.  E.  W.  Hyde,  formerly  of  Cornell  University, 
who  has  with  signal  ability  and  fidelity  assisted  me  in  preparing  the  book  for 
the  press.  In  particular,  the  Examples  were,  almost  without  exception,  made 
by  him. 

Part  Second,  upon  Solid  Geometry,  is  in  preparation,  and  will  be  issued  at  as 
early  a  date  as  circumstances  may  permit. 

---HENRY  T.  EDDY. 
PRINCETON,  NEW  JERSEY, 
August  15, 1874. 


CONTENTS. 


PAGE 

ABBREVIATIONS  AND  SIGNS. — GREEK  ALPHABET 7 

INTRODUCTION 8 

CHAPTER   I. 

Co-ordinates. 

Systems  of  Co-ordinates , 9 

Positive  and  Negative. — Distance  and  Angle 11 

CHAPTER   II. 

The  Point. 

Cartesian  Co-ordinates 14 

Relations  of  Points 18 

Polar  Co-ordinates 22 

Projections 28 

Transformation  of  Co-ordinates 30 

CHAPTER   III. 

The  Right  Line. 

The  Right  Line  in  Cartesian  Co-ordinates 39 

Relations  of  two  or  more  Right  Lines 47 

Perpendicular  and  Direction  Cosines 54  , 

Right  Line  in  Polar  Co-ordinates GO 

Exercises 62 

CHAPTER  IV. 

The  Circle. 

Tangent  and  Normal  of  any  Curve 64 

The  Circle  in  Rectangular  Co-ordinates 65 

The  Tangent  and  Normal  Line 09 

Centre  and  Axes  of  Similitude 73 

Polar  Equation '..  76 

Exercises 76 

CHAPTER   V. 

Equations  and  Loci. 

Definitions 79 

Properties  of  Equations  and  their  Constituents 81 

Symmetry , 91 

5 


CONTENTS. 


CHAPTER  VI. 

The  Parabola.  PAOB 

Definition  of  the  Conic  Sections 93 

The  Parabola  in  Rectangulars 94 

The  Tangent,  Polar  and  Normal  Lines 96 

The  Parabola  in  Polar  Co-ordinates 105 

CHAPTER  VII. 
The  Hyperbola  and  Ellipse. 

The  Hyperbola  and  Ellipse  in  Rectangulars 108 

Tangent,  Polar  and  Normal  Lines 118 

Conjugate  Diameters  and  Eccentric  Angle 128 

Asymptotes  of  the  Hyperbola... 141 

Hyperbola  and  Ellipse  in  Polar  Co-ordinates.... 144 

CHAPTER  VIII. 
General  Equation  of  the  Second  Degree. 

Co-ordinates  of  Centre  and  Direction  of  Axes 146 

Invariants  and  Criteria 150 

CHAPTER  IX. 

Curves  of  the  Third  and  Fourth  Degrees. 

Curves  of  the  Third  Degree 153 

Curves  of  the  Fourth  Degree 159 

CHAPTER   X. 
Higher  Algebraic  Curves. 

Parabolic  and  Hyperbolic  Curves 163 

Exponential  Polygon  and  Approximate  Equations 167 

Multiple  Points 178 

CHAPTER  XI. 

Transcendental  Curves. 

The  Cycloids 180 

Trigonometric  and  Auxiliary  Curves 185 

CHAPTER    XII. 

Spirals  and  Polar  Curves. 

Parabolic  and  Hyperbolic  Spirals 196 

Exponential  Polygon  and  Approximate  Equations 199 

Exercises...  ..  200 


Abbreviations  and  Signs. 


E.  G.  is  used  to  introduce  an  illustrative  example. 

N.  U.  is  used  to  introduce  some  useful  notation  or  convention  adopted. 

A  period  may  signify  multiplied  by,  and  a  colon  may  signify  divided  by.  • 

T  =  8.14159  is  the  semi-circumference  of  the  circle  whose  radius  is  unity. 

.  • .    signifies  therefore,  and  — i.  e.  signifies  that  is. 

OPQ  may  signify  the  angle  OPQ,  etc. 

The  dagger  (f )  signifies  that  the  proposition  to  which  it  is  affixed  may  be  omitted 
if  desirable. 

0  (x,  y)  signifies  some  unknown  function  of  <c  and  y,  and  is  read  phi  function  of 
x  and  y. 

Read  combinations  of  subscripts,  primes  and  powers  as  follows :  Pi,  pe  one; 
P2,petwo;  P/,pe  prime;  P//,pe  second;  P\",pe  one  second;  P'?,  pe  prime 
two;  x'2,  ex  prime  square;  xf,  ex  two  square,  etc. 

&  signifies  the  angle  between  x  and  y  (Art.  12). 


Greek  Alphabet. 


a  alpha. 

*  iota. 

P  rho. 

ft  beta. 

K  kappa. 

a  C  sigma. 

7  gamma. 

A  lambda. 

r  tau. 

6  delta. 

,"  mu. 

v  upsilon. 

e  epsilon. 

v  nu. 

<P  $  phi. 

C  zeta. 

£  xi. 

X  chi. 

T]  eta. 

o  omicron. 

V*  psi. 

0  theta. 

TT  pi. 

w  omega. 

i 

INTRODUCTION. 


Analytic  Geometry  is  distinguished  from  other  geometry*  in  this 
one  particular :  its  investigations  are  conducted  by  means  of  the 
forms  and  symbols  of  algebra. 

Co-ordinate  Geometry  is  that  branch  of  Analytic  Geometry  in 
which  investigations  are  conducted  by  means  of  co-ordinates. 

Co-ordinates  are  quantities  which  determine  position.  Position 
can  be  determined  only  by  reference  to  some  assumed  position. 

Latitude  and  longitude  are  co-ordinates.  They  determine  the  position  of  a 
point  on  the  earth  by  reference  to  the  equator,  and  to  an  assumed  meridian. 

Time  by  the  clock  is  also  a  co-ordinate.  It  determines  a  point  of  time  by 
reference  to  mean  noon. 

*  In  this  treatise  the  elementary  propositions  of  geometry  and  trigonometry 
are  assumed  to  be  already  proven. 


'  '  1  * \   1 I    I 
VKKSIT  V    < 


ANALYTIC   GEOMETRY, 

PAET  FIEST. 

i»r  t  \  i:   <-i  <MI  i  a  K  v 


CHAPTER   I. 

CO-ORDINATES. 

1.  The  position  of  a  point  in  a  plane  may  be  determined  by 
co-ordinates  of  many  different  kinds.     A  few  of  these  systems 
are  as  follows. 

2.  Focal  co-ordinates. — From  the  given  or  assumed  points  Fl 
and  F2t  let  the  point  Pl  be  at  the  distances  TI  and  r2  respectively. 
Then  rL  and  r2  are  the/oca? 

co-ordinates  of  PI.     It  is  to 

be  noticed  that  TI  and  r2  are 

also   the    co-ordinates   of   a 

second  point  P2.     The  two 

triangles  P&F^  and  P^F* 

are    evidently    equal,    since 

their  corresponding  sides  are 

equal,  and  hence  the  points 

PI  and  P2   are   symmetrically  situated  with  reference  to  the 

right  line  F^j  joining  the  points  FiF2.      The  points  FL  and 

F2  are    called  foci,  and  the   right  line  joining  them   is   called 

an  axis.     This  system  of  co-ordinates  is  at  present  of  limited 

application. 


10 


CO-ORDINATES. 


3.  Angular    co-ordinates. — From 
the  given  or  assumed  points  Oi0203 
let  the  right  lines  to  P  contain  the 
angles  dl  and  02',   then  6\  and  62 
are  the  angular  co-ordinates  of  P. 
This  system  is  used  in  the  topo- 
graphic work  of  the  United  States 
Coast  Survey. 

4.  Polar  co-ordinates. — From  the  given  or  assumed  point  0 
upon  the  given  or  assumed 

line  OX,  let  OP  have  a 
length  p,  and  make  an 
angle  6  with  OX',  then  p 
and  6  are  the  polar  co- 
ordinates of  P.  This  sys- 
tem of  co-ordinates  is  much 
used  in  ordinary  analytic 
work. 


0 


5.  Linear  co-ordinates. — If  from  the  point  Pl  we  let  fall  the 
perpendicular  APi  upon  the  given  or  assumed  line  D,  and  let  it 
have  the  length  d,  and  also 
draw  from  the  given  or  assumed 
point  F  the  right  line  FPl 
with  the  length  p ;  then  p  and 
d  are  the  linear  co-ordinates  of 
P!.  The  line  D  is  called  a 
directrix,  and  the  point  F  a 
focus.  If  a  line  be  drawn 
through  F  perpendicular  to 
D,  it  will  be  an  axis.  It 
will  be  observed  that  p  and  £ 
are  also  the  co-ordinates  of 

a  second  point  P2,  and  that  PI  and  P2  are  symmetrically  situated 
with  reference  to  this  axis.  This  system  of  co-ordinates  is 
employed  to  some  extent  in  this  work,  in  the  treatment  of  the 
conic. 


POSITIVE  AND  NEGATIVE. 


11 


6.  Bilinear  OP  Cartesian  co-ordinates. — From  P  draw  two  lines 
PB  and  PA  respectively 
parallel  to  two  given  or 
assumed  right  lines  OX 
and  0  Y;  then  OA  =  x  and 
OB  =  y  are  the  Cartesian 
co-ordinates  of  P. 


When  XOY=90°,  the 
system  is  called  rectangular, 
and  is  the  system  princi- 
pally employed  in  this 
treatise. 


<o 


7.  Trilinear  co-ordinates. — From  P  let  fall  three  perpendiculars 
aftf  upon  the  sides  of  a  given  or 
assumed  triangle  Oi0203;  then  a/9f 
are  the  trilinear  co-ordinates  of  P. 
This  system,  together  with  its  recip- 
rocal system,  tangential  co-ordinates, 
is  used  in  modern  analytic  investi- 
gations. 


POSITIVE  AND   NEGATIVE. 

8.  Motion  is  of  two  kinds,  that  of  translation,  and  that  of 
rotation. 

The  distance  between  two  points  is  the  amount  which  a  point 
must  move  in  passing  from  one  of  the  points  to  the  other. 

If  distance,  or  motion  of  translation  of  a  point  in  one  direction 
be  assumed  to  be  positive,  then  motion  in  the  opposite  direction 
will  be  negative. 


Distance  or  length  is  measured  in  feet,  inches,  metres,  or  some  other  arbitrary 
unit. 


12 


CO-ORDINATES. 


0.  If  the  distance  from  0  to  A  is  OA,  and  that  from  A  to  0  is 
;  then  OA  =  -AO. 


If  £0  =  0.4, 

then  OA=  -  OB,  and  AB  =  -  BA. 

Also  OA  +  A  C=  OA-CA  =  OC, 

or  BO  +  OD  =BO  -DO  =BD, 

and  OD  +  DC+CA=-D\ 

.-.  0.1=0  F+FFrfF^f... 
in  which  F,  FI,  T^,  etc.,  are  any  points  whatever  in  the  line  AO. 

10.  The  angle  between  two  lines  is  the  amount  which  a  line 
must  turn  in  passing  from  one  of  the  lines  to  the  other. 

If  the  angle  or  rotation  of  a  line  in  one  direction  be  assumed  to 
be  positive,  then  motion  in  the  opposite  direction  will  be  negative. 

Angle  or  rotation  is  measured  in  degrees,  grades  or  arbitrary  fractions  of  an 
entire  rotation.  The  Greek  letter  TT  is  commonly  used  to  designate  the  length  of 
the  semicircumference  of  the  circle  whose  radius  is  unity. 

•  11.  If  the  angle  from  OX 
to  OA  is  XOA,  and  that  from 
OA  to  OX  is  A  OX,  then  XOA 
-AOX.  If  BOX=XOAt 
then  XOA  =  -XOB,  and  BOA 
=  —AOB.  The  direction  indi- 
cated by  the  arrow  is  the 
direction  which  will  be  con- 
sidered positive. 

12.  N.  B. — If  the  direction  OX  be  called  x,  for  convenience, 
and  OA,  plt  then  the  angle  XOA  may  be  written  £»*  an(J  will  be 
read,  "  the  angle  between  x  and  plf"  . 


*  This  must  be  carefully  distinguished  from  a  fractional  expression.     The  lower 
letter  is  written  first  and  read  first. 


POSITIVE  AND  NEGATIVE.  13 

Then  ^OJTwill  be  written  x' 

"\ 

Since  XOA  =  -AOX,  we  have  £  =  -pi- 
Also  P*  +  Pi  =  P3-P*  =  Pi. 

X         ps       X         pl       X' 

or  P*  +  Pi=Pi.9 

P2        P3        P*' 

P-+P-+P'  +  Pi=Pi 

X+P>P*Pr        X' 

in  which  />„,  pm  and  />r  have  any  directions  whatever.  Therefore 
the  equation  a  +  ^  +  £  +  +^1=^1  expresses  the  final  angle 
between  x  and  a  line  turning  from  x  successively  to  a  /?  7-  £  and  />L. 

It  is  to  be  noticed  that  all  parallels  have  the  same  direction,  so  that 
any  line  has  the  same  direction  as  a  parallel  to  it  through  0;  from 
which  it  follows  that  the  sum  of  the  exterior  angles  of  any  convex 
polygon  is  equal  to  360°,  as  is  also  proved  in  geometry. 

It  is  also  to  be  noticed  that  any  multiple  of  27r  =  3600  may  be 
neglected,  since  a  complete  revolution  does  not  change  the  position  of  a 

line.     Also  that  ~~ *  =  180°. 

JC 

13.  Since  yx=—  *,  in  which  x  and  y  have  any  directions  what- 
ever, we  have  by  trigonometry : 


tan  |  =  tan  (-*)  =  - tan* 

cot  |  =  cot  (-*)= -cot* 

sec  |  =  sec  (-*)  =  +  sec  * 

cosec  y  —  cosec  —?.  =  —  cosec 


^^  CHAPTER   II. 

THE   POINT. 

CARTESIAN    CO-ORDINATES. 

14.  Abscissa,  or  x.     The  distance  OAl=BlPl  is  called  the 
abscissa  of  P1;  or  the  x  co-ordinate  of  Px,  or  simply  the  #  of  PI. 


A,     / 


When  this  co-ordinate  is  measured  to  the  right  of   0,  it  is 
usually  reckoned  positive,  and  when  measured  to  the  left,  negative. 

15.  Ordinate,  or  y.      The   distance  OBl  =  AlPl  is   called  the 
ordinate  of  Plt  or  the  y  co-ordinate,  or  simply  the  y  of  Pl4 

It  is  usual  to  consider  y  positive  above,  and  negative  below  0; 
e.  g.,  for  the  point  P3,  x  and  y  are  both  negative. 

16.  Axes. — The  lines  OX  and  0  Fare  called  the  axes  of  reference, 
or  the  co-ordinate  axes,  or  simply  the  axes  of  #  and  y. 

The  axis  of  re  is  usually  taken  horizontal. 

14 


ORIGIN  AND  AXES.  15 


17.  Origin. — The  point  0,  in  which  the  axes  of  x  and  y  inter- 
sect, is  called  the  origin  of  co-ordinates,  or  simply  the  origin. 

+     + 
The  angle  XOYis  called  the  first  angle. 

"      "      XOY    "         "      second" 

"      "      XOY    "         "      third    " 

+    - 
"       "      XOY    "         "     fourth  " 

For  a  point  situated 

in  the  first  angle,  x  is  +  and  y  is  +. 
in  the  second  angle,  x  is  —  and  y  is  +. 
in  the  third  angle,  x  is  —  and  y  is  — . 
in  the  fourth  angle,  x  is  +  and  y  is  — . 

18.  Oblique  Axes. — When  the  angle  between  the  axes  of  x  and 
y  is  not  90°,  the  system  is  called  oblique. 

+      4* 

It  will  be  convenient  to  use  at  =  XO  Fto  denote  this  angle. 

Rectangular  Axes. — When  o  =  90°,  the  system  is  called 
rectangular. 

19.  N.  B. — We  shall  use  x  and  y  to  denote  the  co-ordinates  of  any 
point  P  with  reference  to  the  axes  OX  and  0  Y, 

We  shall  use  x1  and  2/  to  denote  the  co-ordinates  of  any  point  P  with 
reference  to  other  axes,  OX'  and  O  Z',  and  a;"  and  y"  with  reference 
to  0"X"  and  0"  Y",  etc.,  etc. 

We  shall  also  for  convenience  use  xl  and  yv  to  denote  co-ordinates  of 
some  particular  point  jP1?  i.  e.,  xl  and  ^  are  particular  values  of  x  and 
y.  Also  #2  and  y2,  for  the  point  jP2>  are  other  particular  values  of  x 
and  y,  etc.,  etc. 

Similarly,  a?/  and  y/,  or  #/  and  y/  are  particular  values  of  of  and  £/. 


16  THE  POINT. 


Proposition  1. 

20.  TJieorem.—The  equations 

x  =  Xi    and  y=y\ 

represent  a  point ;  in  which  x  and  y  may  be  the  bilinear 
co-ordinates  of  any  point,  and  Xi  and  yl  are  their  values 
for  this  particular  point. 

For  a  single  point  has  position  only,  and  its  position  is  com- 
pletely determined  by  these  equations. 

N.JB. — We  shall  frequently  speak  of  the  point   (x^y^)t   or  ( — 2, 3), 
etc.,  meaning  the  point  whose  co-ordinates  are 

x  =  x^  y  —  y\,  or#  =  — #,    2/  =  #,  etc. 

21.  Corollary. —  The  equations  x  =  0  and  y  =  0  represent  the  origin. 

22.  Examples. — Locate  the  points  represented  by  the  following 
equations,  both  in  rectangular  and  oblique  co-ordinates,  using  first  \  in., 
then  i  in.,  as  the  unit  of  measure. 

(1.)  *  =  *,  y=6. 

(2.)          x  =  -l,  y  =  3.5. 


(5.) 
(6.) 
(7.) 
(8.) 


TRANSFORMATION. 


Proposition  2. 

23.  Theorem.— The  equations 

I   "'<J/ 

are  the  equations  by  which  any  point  is  referred  to  a  new 
system  of  axes  parallel  to  the  primitive  system;  in  which 
x  and  y  are  the  primitive  co-ordinates  of  any  point,  x' 
and  y'  the  new  co-ordinates  of  the  same  point,  and  x0 
and  2/0  are  the  co-ordinates  of  the  new  origin. 

The  equations  are  evidently  true  from  inspection  of  the  figure. 
For,  OA  =  x  =  xQ  -f  x'j  and 


24.  Schol.-Ey  Art.  23, 

X'=X—XQ, 
and   y'=y-y» 


- 


It  is  evident  that  x0  and  y0  are  positive  when  measured  from 
the  origin  0,  but  negative  when  measured  from  0', 

.  •  .     XQ  =  -#</>  and  y0  =  —yj- 
Substituting  in  the  previous  equations,  we  have 
xf  =  x0'  -f  a?,  and  y'  =  y'^y, 

.'.     x  =  xf—  xj,  and  y  =  y'—  y0'. 
These  axes  may  be  either  oblique  or  rectangular. 

The  above  change  of  axes  of  reference  is  a  case  of  transformation 
of  co-ordinates.  This  change  will  evidently  not  affect  the  relations  of 
the  point  to  any  other  points  or  lines  than  the  origin  and  axes. 


18 


THE  POINT. 


25.  Examples. — Construct  the  axes  and  points  in  the  following 
examples  (see  Art.  18)  : 


(4.) 
y  ==— 


9    /  .J     (j>  n  /    G)  fy*      Q 

X   ~      — fa  X   — O,  y    — ^,  «//0  —  ^> 

y  =  5,  y  =  — £,  2/o  =  — -4,  ij<>  =  —  2, 

a>  =  45°.  a>  =  90°.  a>  =  120°.  o>  =  ^°. 

(5.)  The  three  vertices  of  a  triangle  are  the  points,  (2,  1\  (3,  —  4), 
(—  3,  —2),  and  the  co-ordinates  of  the  new  origin  are  XQ  =  5,  y0  =  —  4 ; 
construct  the  triangle,  and  find  the  new  co-ordinates  of  the  vertices, 
when  10  =  90°  :  also  when  at  =  120°. 


Proposition  3. 

26.  Theorem.— The  equation 


expresses  the  distance  between  two  points ;  in  which  r  is 
the  distance,  and  (xi,yi)  and  (z2,  y2)  are  the  rectangular 
co-ordinates  of  the  points. 

Since  the  square  on  the  hypote- 
nuse is  equal  to  the  sura  of  the 
squares  on  the  two  sides  of  a  right- 
angled  triangle,  the  equation  is 
evidently  true  from  inspection  of 
the  figure.  For 

P^=  IFD  +  P^D  =  (OC-  OA}+  (CP2  -  CD)] 


27.  Cor.— If  xl  =  0,  and  y^  =  0  (Art.  21),  then  r  =|/^22  +  2/22. 

It  is  to  be  noticed  that  this  proposition  is  general,  though  proved 
only  in  the  first  angle. 


DISTANCE.    AREA.  19 


28.  Schol. — The  expression 


r  =  -\/(xt  —  xtf  +  (y2  —  yO2, 
when  transformed  to  new  parallel  axes  by  the  equations  (Art.  23) 


is          r  =  i/(x2'  —  x^y  +  (y/  —  y/)2. 
For         a?,  -  a*  =  [a*  +  ara'  -  (x9  +  a?/)]  =*2'  -  a?/, 
and  similarly       y2  —  yi  =  y/  —  y/. 

It  is  to  be  noticed  this  transformation  does  not  affect  the  relation  of  the  points 
to  each  other. 

29.  Examples. — Find  the  distances  between  the  following  points : 

(t)         (3,  —I),     and  (8,5).  Ans.  7.81. 

(2.)         (—2,  —4\     and  (1,  —2). 
(&)         (3,7),     and  (—  3,  5). 

(4.)  Refer  the  points  in  Ex.  1  to  new  parallel  axes,  the  co-ordinates 
of  the  new  origin  being  x0  =  6,  y0  =  2. 

(5).  In  Ex.  3  refer  to  new  parallel  axes,  making  x0  =  3  and  y0  =  7. 

30.  Exercise. — Prove  that  when  the  axes  are  oblique, 

r  =  i/(#2  —  xtf  -I-  (y3  —  yO2  +  2  (x2  —  xj  (y2  —  yj  cos  w. 

Proposition  4.f 

31.  Theorem.— The  equation 

expresses  twice  the  area  of  the  triangle  whose  vertices  are 
the  three  points,  of  which  the  rectangular  co-ordinates  are 
(#1, 2/i),  (»*,  2/2),  (a?s»  2/s)>  <*«•<£  0/  which  the  area  is  t. 


f  All  propositions  or  other  divisions  of  the  book  marked  (  f )  may  be  omitted 
on  first  reading. 


20 


THE  POINT. 


For,  t  = 


ljr  a.2  b.2) 


-fe-^i)  (y3+yi) 


0         tti 


0,3 


0,2     X 


And  by  multiplying  out  and  canceling  we  obtain, 


32.  Cor.  —  In  the  same  manner  we  can  obtain  the  area  of  any  poly- 
gon.    The  area  of  a  quadrilateral  =  q  is  given  by  the  equation, 

*  —  #  0 


33.  Schol.  1.  —  If  the  preceding  points  be   referred  to   new   axes 
parallel  to  the  primitive,  we  shall  have  (Art.  23), 

yi  (*,'  -  ti)  +  yi  (?l  -  x!)  +  yl  W  -  x{)  =  ±  to, 

since  all  the  terms  that  contain  XQ  and  y0  will  cancel. 

The  relation  between  the  points  is  unchanged  by  the  transformation. 


34.  Schol.  2. — It  is  useful  to  notice 
the  cyclic  symmetry  of  the  above  ex- 
pression— that  is,  that  the  subscripts 
follow  around  in  the  same  cyclic  order, 
viz.,  123,  231,  312. 


35.  Examples. — Find  the  area  of  the  surfaces  inclosed  by  right 
lines  joining  the  points  whose  co-ordinates  are  given  in  the  following 
examples : 

(1.)         (2,3),         (4,1)  and  (5,  6).  Ans.6. 

(2.)        (-5,1),    (-2, -3),    (4,6).  An*.  26$. 

(3.)        (-4,1),     (6,3),  &-1),      (-3,  -2).     Am.  51. 


THREE  POINTS  ON  A  LINE. 


21 


36.  Exercise.  —  Prove  that  when  the  axes  are  oblique, 

[y\  fe  —  #3)  +  2/2  (^s  —  #0  +  3/3  (x\  —  #2)]  sin  w  —  ± 


Proposition  5. 

37.  Theorem.—  Tlie  equations 
n,  x, 


__ 


express  the  position  of  a  point  dividing  the  distance  between 
two  points  in  a  given  ratio;  in  which  n\  :  n2  is  the  ratio, 
and  (x,  y}  are  the  co-ordinates  of  the  point  dividing  the 
distance  between  (x^y^)  and  (xt,yt)  in  the  given  ratio. 

For,letOa=z,     Oal=x1    and 
By  similar  triangles  we  have 
bid  CD  :  :  b,P  :  Pb2, 


or 


Similarly  y  = 

J   ^ 


X 


a     a 


38.  Schol.  2.  —  If  the  preceding  points  be  referred  to  new  axes  paral 
lel  to  the  primitive,  since  (Art.  23), 

x  —xv  =  x0  +  x'  —  (x0  +  Xi)  =  xf  —  or/,  etc.  ; 
.  •  .    of  —  Xi  :  x2'  —  x1  :  :  n^  :  n^  ; 


d 


ni  + 


+ 


This  transformation  does  not  change  the  mutual  relation  of  the  three  points. 


22  THE  POINT. 


39.  Schol.  2.  —  Were  the  point  not  situated  between  (xlt  ?A)  and 
(#2,  y*)>  but  on  either  sidb  of  these  points,  n?  would  be  negative,  and 
the  equations  would  become 

d 


40.  Examples.  —  (1:)   Find   the   co-ordinates   of  the   point  which 
divides  in  the  proportion  of  3  to  5  the  line  joining  the  points 


and   «  >       Ans.  <          38 

\y=i. 

Given  a  line  joining  two  points  (5,  —  #)  and  (—8,  7),  to  find 
the  co-ordinates  of  a  point  on  the  line  not  between  the  given  points, 
which  shall  divide  the  line  in  the  proportion  of  1  to  7. 


41.  Exercise.  —  Prove  this  proposition   also  as  a  case  of  Prop.  4, 
when  t  =  0. 


POLAR    CO-ORDINATES. 

42.  Radius   Vector. — The  distance    OP  is   called  the  radius 
vector,  or  p  co-ordinate,    or  simply 

the  p  of  P. 

It  is  usual  to  consider  p  as  positive 
when  there  is  no  reason  for  taking 
it  negative. 

43.  Variable   Angle. — The    angle 
XOP  is  called  the  variable  angle, 
or  6  co-ordinate,  or  simply  the  6 
of  P. 

It  is  usual  to  measure  this  angle  from  XO  around  in  a  direction 
opposite  to  the  motion  of  the  hands  of  a  watch  for  positive  rotation,  and 
with  the  hands  for  negative  rotation. 

It  is  often  convenient  to  denote  the  angle  between  x  and  /?,  as  p  =0, 
which  must  be  carefully  distinguished  from  a  ratio  or  fraction.  (Art.  12.) 


POLAR  CO-ORDINATES.  23 

44.  Pole. — The  point  0  in  which  all  the  radii  vectores  inter- 
sect is  called  the  pole.     The  pole  is  the  origin  of  distance. 

Initial  Line. — The  line  OX  from  which  the  variable  angle  is 
measured  is  called  the  initial  line.  The  initial  line  is  the  origin 
of  direction. 

45.  N.  B. — "We  shall  use  p  and  0  to  denote  general  values  of  the 
polar  co-ordinates  referring  to  a  primitive  pole  and  initial  line,  and  p' 
and  tf  referring  to  a  new  pole  and  initial  line.     We  shall  also  use  f)l 
and  #i,  PZ  and  02,  etc.,  to  denote  restricted  values.     (Compare  Art.  19.) 


Proposition  6. 

46.  Theorem.— The  equations 

p=pl  and  0  =  0! 

represent  a  point;  in  which  p  and  0  may  be  the  polar  co- 
ordinates of  any  point,  and  plt  Ol,  are  their  values  for  this 
particular  point. 

For,  a  single  point  has  position  only,  and  its  position  is  com- 
pletely determined  by  these  equations. 

N.  U. — We  shall  use  (pl}  0J,  or  (5,  -),  etc.,  to  indicate  the  point  whose 
co-ordinates  are  p=pi,  0  =  0^  or  p  =  5,  0  =  ^,  etc. 

47.  Cor. — The  equation   p  =  0  represents  the  pole,  and  the  equa- 
tion  0  =  0,   or  0  =  -,   or  0  =  2  TT,    etc.,  represents  the  initial  line. 

48.  Examples. — Locate  the  points  whose  co-ordinates  are  given 
below. 

(1.)  p=2,         0  =  1. 


(2.)          P  =  -l,     0  =  45°. 

e 
(3.)          P=     »,     0  =  2^. 


24 


THE  POINT. 


Proposition  7. 

49.  Theorem.—  The  equation 


is  that  by  which  any  point  is  referred  to  a  new  initial  line 
through  the  pole;  in  which  0o  is  the  angle  between  the  new 
and  the  primitive  initial  line. 

From  inspection  of  the  figure, 

P    —  X'    _L  P  >r>  A  —  ft      -4-   ft' 


Observe  that  the  transformation  of 
this  one  of  the  polar  co-ordinates  of  any 
point,  will  not  change  its  relations  to 
anything  except  the  initial  line  itself. 


0 


Proposition  8. 
50.  Ttieorem.—The  equation 


f  —  1/Y'i2  +  P/  ~  Mf'i  ('2  cos  (#1  —  #2) 

expresses  the  distance  between  two  points ;  in  which  r  is 
the  distance  between  the  two  points  whose  polar  co-ordi- 
nates are  (/>i,  0i)  and  G°2,  #2). 

If        a}  02  =  r,    Oai  =  pl}    Oa2  =  p2)  ®i 

and     £*  =  6l  —  02,  we  have,  by  trig., 


(Compare  Art.  30.) 

51.  Cor.  —  If  one  of  the  points,  say 
a2,  be  at  the  origin,  r  =  p. 


52.  Schol.  —  If  the  initial  line  be  changed  to  coincide  with  p2t 
i.e.,          00  =  02,  .'.      0i  -  02  =  00  +  */  -  ^o, 


-/>!/>,  cos 


AREA   OF  TRIANGLE. 


25 


53.  Examples.  —  Find  the  distances  between  the  points  whose  co- 
ordinates are  as  follows  : 


(!•) 

(2.) 
(3.) 

(4.) 


Am.  r  =  6.08 


t 


=8,0  =  350°,  and  p  =  6,  0  =  80°. 


Ans.  r  =  5. 


9  ' 


Ans.  r  =  10. 


Ans.  r  =  8.72 


Proposition  9.f 
54.  Tlieorem.—The  equation 
p1  p2  sin  (0i  —  02)  +  p2  ps  sin  (02  —  03)  +  p3  pl  sin  (03  —  0j)  =  ±  ££ 

expresses  twice  the  area  of  any  triangle,  when  t  is  its  area, 
and  the  polar  co-ordinates  of  its  vertices  are  (P\,  0\),  (f>*,  0*}, 
and  G°3,  0s). 

The  triangle  123  is  equal  to 

012  +  023  +  031, 
and  by  trig,  the  area  of  012  is 

01X02X  sin  102     p,  p2  sin  (0!  -  02) 

and  similarly  for  the  other  triangles, 

.  • .   ptp2  sin  (0!  -  02)  +  p2  ps  sin  (02  -  03)  +  p3  pl  sin  (03-  0^  =  ±  2t. 

This  expression  is  equally  true  when  the  pole  is  not  within  the 
triangle. 

Observe  the  cyclic  Bymmetry  in  the  above  expression. 


26  THE  POINT. 


55.  Cor.  —  In  the  same  manner  we  may  derive  an  expression  for  the 
area  of  any  polygon. 

56.  Examples.  —  Find  the  areas  included  within  right  lines  joining 
the  points  whose  co-ordinates  are  given  below. 

(1.)     (5,  10°),     (2,  100°),     (3,  200°).  Ans.  t  =  9.26 

<2-)    *•   *        '•• 


Proposition 

57.  TJieorem.—The  equation 

2  - 


(pf  -  pf)  (pf  -  pf)  +  rfpf  (r?  +  n2  -  rf) 
-  rf  C«32  -  P?)  (!>?  ~  P/)  +  rf  tf  (r?  +  rf  -  rf)  =  rf  r}  r," 

expresses  the  relation  between  the  length  of  the  sides  and 
diagonals  in  any  quadrilateral  —  i.  e.,  the  six  distances  be- 
tween any  four  points. 

For  identically, 

(03-dJ  +  (Oi-02)  =  03-02,    or«  +  /9  =  r, 

if      0,-dl  =  a)       flx-02^/3    and  03-02  =  ;- 

By  trig,   sin  a  cos  /3  +  cos  a  sin  /9  =  sin  f 
Squaring, 

sin2  a  cos2  /9  +  2  sin  a  sin  /?  cos  «  cos  /9  +  cos2  a  sin2  /?  =  sin2  /• 

.  •  .     (-?  —  cos2  «)  cos2  j9  +  ^  sin  a   sin  /9  cos  a   cos  /9 
+  cos2  a   sin2  ft==l  —  cos2  ^, 

.  *  .     cos2  a  (cos2  p  +  sin2  /?)  +  cos2  /9  4-cos2  y 

—  2  cos  a   cos  /9  (cos  a   cos  /?  —  sin  a  sin  /9)  ==^, 

.  '  .    cos2  a  +  cos2  /9  +  cos2  p  —  2  cos  a  cos  ^  cos  f  =  l  .....  (a.) 


FOUR  POINTS. 


27 


By  Art.  50, 


ll'2 


o  x 

Substituting  these  values  in  eq.  (a.}}  we  have  the  above  result. 


Proposition  H. 

58.  Theorem. — The  equation 

AP  .  QB     sin  AOP  sin  QOB 
AQ  . 


=  a  constant 


expresses  the  relation  of  the  four  distances  between  the  four 
points  in  which  any  line  intersects  a  pencil  of  four  rays. 

For,  if  p  =  length  of  the  perpendicular 
let  fall  from  0  upon  AB,  we  obtain  the 
following  expressions  for  double  areas  of 
triangles. 

p.  AP  =  OA.  OP  sin  AOP  .  . 
p.  QB  =  OQ  .  OB  sin  QOB  .  . 


p.AQ  =  OA  .  OQsmAOQ 
p.PB  =  OP .  OBsmPOB 


(d.)       o 


The  product  of  equations  (a.)  and  (6.),  divided  by  the  product 
of  equations  (c.)  and  (d.),  gives  the  above  result,  which  is  the  same 
for  every  line  intersecting  the  pencil. 


28  THE  POINT. 


59.  Scfiol.—  AP.  QB  :  AQ  .  PB 

is  called  the  anharmonic  ratio  of  the  pencil  0-APQB. 
The  ratios  AP  .  QB  :  AB  .  PQ 

and  AB.QP-AQ.PB 

are  also  of  constant  value,  as  may  be  proved  in  a  similar  manner. 

PROJECTIONS. 

60.  Projection  of  a  Point.  —  The  foot  of  a  perpendicular  let  fall 
from  any  point  upon  a  line  is  the  orthogonal  (i.e.,  rectangular) 
projection  of  the  point  upon  the  line.     Similarly  the  oblique  pro- 
jection of  a  point  may  be  obtained,  but  the  orthogonal  projection 
will  always  be  understood  unless  otherwise  stated. 

61.  Projection  of  a  Distance.  —  The  distance  between  the  pro- 
jections of  two  points  is  the  projection  of  the  distance  between 
the  points. 

Proposition  12. 

62.  Theorem.—  The  equations 

x2  —  x1=r  cos  £,    and  y2  —  y\  =  r  sin  Tx 

express  the  projection  of  the  distance  between  two  points 
upon  the  rectangular  axes  of  x  and  y;  in  ivhich  r  is  the 
distance  between  the  two  points 


For,  if  im  =  x2  —  xl,  and71?  =  r, 
by  trig.        xz  —  x1  =  r  cos 


x2  —  Xi  —  r  cos  Tx 


Similarly  y2-  yl  =  r  cos  (90Q  -  ^ 

.*.     by  trig.  y2 — y1  =  rsin![/  ^  ^2  -X 

The  given  distance  is  the  hypotenuse  of  a  right-angled  triangle, 
and  its  projections  on  the  axes  of  x  and  y  are  respectively  parallel 
and  equal  to  the  base  and  perpendicular. 


PROJECTIONS. 


29 


63.  Cor.  —  If  the  point  1  coincides  with  0, 


x-2  =  r  cos  r  ,     and  y-2  =  r  sin  r. 


64.  Example.  —  Find  the  projections  on  the  axes  of  x  and  y  of  the 

distance  between  two  points,  when 


=  £,   and;*  = 

JC 


Ans.  #2  —  #1  =  •£,  and  y^  —  yl  =  3. 


Proposition  13. 

65.  Theorem.— The  equations 

TI  cos  !}  =  r2  cos !?  -f  rB  cos 


sn 


r2  sin 


sn 


express  the  fact  that  the  projection  of  one  side  of  any 
triangle  upon  any  lines,  as  upon  the  rectangular  axes  of 
x  and  y,  is  equal  to  the  sum  of  the  projections  of  the  two 
remaining  sides  upon  the  same  line  ;  in  which  rltrt  and  r3 
are  the  three  sides  of  the  triangle. 


For, 


Also, 


=  02  a:  +  a\  a3 


cos     =  r2  cos  T  +  r3  cos 


&s  =  &2&i  +  &i&8;  (Art.  9.) 
sin  £  =  r2  sin  £2  +  r3  sin  J, 


«3  X 


66.  Cor. — Since  r2  might  be  the  third  side  of  a  new  triangle,  etc.,  it 
follows  that  the  sum  of  the  projections  upon  the  axis  of  ar,  of  any  broken 
lines  leading  from  2  to  8,  is  equal  to  the  projection  of  the  distance  2  8 
upon  the  axis  of  x.  By  the  word  sum  is  to  be  understood  algebraic 
sum,  since  the  projections  of  any  points  falling  without  a^  a$,  would 
give  us  negative  distances.  (Art.  9.) 


30 


THE  POINT. 


67.  Example. — Show  the  truth  of  the  preceding  proposition  numeri- 
cally by  applying  the  formula  to  the  following  data.  Co-ordinates  of 
(!),*  =  $,  y  =  2;of(2),x  =  6,  y  =  l;  of  (3),  a?  =  0,  y  =  6; 


r\_ 


25"         = 


=  161°  34'. 


TRANSFORMATION. 


68.  The  Transformation  of  the  co-ordinates  of  any  point  is  the 
reference  of  the  point  to  a  new  system  of  co-ordinates. 


•/ 


N.  B.  Review  Arts.  12,  IS,  18  and  19. 

Proposition  14. 

69.  Theorem.— The  equations 

x  sin  *  =  x'  sin  ^'  +  y'  sin  ^ 
y  sin  yx  =  a;'  sin  *'  +  y '  sin  % 

are  the  equations  of  transformation  of  any  point  from  a 
primitive  system  of  oblique  bilinear  co-ordinates  to  a  new 
system,  also  oblique,  the  origin  remaining  the  same;  in 
which  x  and  y  are  the  primitive  co-ordinates  of  the  point, 
and  x'  and  \f  the  new  co-ordinates  of  the  same  point. 

Project  the  figure  OAPA'  upon  PD 
drawn  perpendicular  to  the  axis  of  x. 
Then,  by  Art.  65, 


Similarly,  if  a  perpendicular  be  let 
fall  upon  the  axis  of  y,  and  we  can 
prove  that 

x  sin  xy  =  x'  sin  *'  +  y'  sin  jj'. 
Notice  the  symmetry  of  these  formulae. 


A     D      X 


TRANSFORMA  T10N. 


31 


70.  Examples.— (1.)  Refer  the  point  x  =  3,y  =  5,  when  ^  =  120°, 

to  new  axes  in  which  ^/  =  £0°,  and  *  =30°,  the  origin  remaining  the 
same.  Ans.  *'  -  '0.577,  y'  ==  4.0^ 

(2.)  Refer  the  point  x  =  2,  y  =  —  5,  when  ^  =  £0°,  to  new  axes  such 
that  yx  —  120°,  and  yx,  =80°,  the  origin  remaining  the  same. 

Ans.  x'  =  -  2.64, 2/  =  -  8.045 


Proposition  IS. 
71.  TJieorem.—The  equations 

x  =  x'  cos  ®  4-  y'  cos  & 

*c>  */  *O 

y  =y'  sin  ^  +  x1  sin  * 

transform  from  rectangular  to  oblique  axes,  the  origin 
remaining  the  same. 

Y     /r 

By  projections 

x  =  x'  cos  x  +  y'  cos  ^ 

«6  ^  •«/ 


72.  Cor. — If  the  axes  of  x  and  x'  coincide, 
then  ?'  =  0,         .'.    ^ 


If  the  axes  of   y  and  y  coincide,  then 


y,=o, 


=  a;  cos 


32 


THE  POINT. 


73.  Examples.— (1.)  Eefer  the  points  (5,  £),  (-  4, 0)  when  £  =  00°, 

to  new  axes  for  which  ^  =  $0°,  and  #'  =  6(9°. 
x  x 

Ans.  For  first  point,       x'  =  0.196,  tf  =  5.6(5 
For  second  point,  x'  =  —  8.93,  y'  =  746 

(2.)  Refer  the  point  (3,  —£)  when  ^  =  00°  to  new  axes  for  which 
x' 


Ans.  x'  =  -  0.732,  tf  =  —  6.175 

74.  Exercise. — Prove  the  equations  of  Art.  71  directly  from  Art.  69. 

Proposition  16. 

75.  Theorem.— The  equations 

x  sin  xy  =  xf  sin  *'  +  y'  cos  *' 
y  sin  y  =  xf  sin  *  +  y'  cos  *' 

•^  *</  *^  U/ 

transform  from  oblique  to  rectangular  axes,   the  origin 
remaining  the  same. 

For  in  the  equations  of  Art.  69,        \^ 
let     $  =  90°,  then  (Art,  12), 


\y 

Y' 

Y 

I 

\ 

1  1\ 

\ 

/yi  \ 

\ 

^\y 

,.-'-"'" 

0 

*\~- 

^\    x 

^^^^ 

. ' .     (by  trig.)  sin  y  =sin  (^  +00°j  —cos 

and  sin  &,'  =  sin  fi£'+  90°\  =  cos  *'. 


.Y 


Substitute  these  values,  and  we  obtain  the  equations  given 
above. 


TEA  NSFORMA  TION. 


33 


76.  Cor.  1. — If  the  axes  of  x  and  x'  coincide,  then  *  =  0,     and 
x'  =x 

y     y' 


y      >.     \<4 

>/..     " 

77.  Cor.  2. — If  the  axes  of  y  and  ?/  coincide,  ( ^ 

/v/          v> 

" 

, 


then 


=  90°. 


y  sin     =  a;'  sin 


cos 


78.  Example. — Refer  the  point  (—3,  6)  when  ^  =  60°  to  new  rec- 
tangular axes,  such  that  ^  =  70°.  Ans.  x'  •=  —  1.719,  ?/  =  4-^03 

79.  Exercise. — Prove  the  equations   of  Art.  75   from   the   figure 
directly. 

Proposition  IV. 

80.  Theorem.— The  equations 

x  =  x'  cos  -J  —  y'  sin  ^ 


y—x'  sin  ~  +?/  cos  ~ 

transform  from  rectangular  tj  new  rectangular  axes,  the 
origin  being  unchanged. 

For,  in  the  equations  of  Art.  69, 
let  y=y' 


then  =     += 

.  • .      sin  |'  =  sin  (^  +  90°\  =  cos 
Also         f^^^-^ 
.'.      sin  ^=  sin  (^-00°)=  -c 
Moreover,     sin  ^  =  —  sin  *  =  7, 
and  sin'=  sin     . 


THE  POINT. 


Substituting  these  values  in  the  equations  of  Art.  69,  we  have, 
x  —  xf  cos  xx  —  y '  sin  xx 
y  =  xf  sin  *'  +  y'  cos  *'. 

This  is  the  most  common  transformation,  and  may  also  be 
written 

x  =  xf  cos  6  —  y'  sin  6 
y  =  #'  sin  6  +  y'  cos  0, 

6   being   the   angle   through   which   the    rectangular  axes   are 
revolved. 

81.  Example. — Refer  the  point  (5,  —  2)  to  new  rectangular  axes 
so  that  we  shall  have  6  =  30°. 

Ans.  x'  =  3.33,  y'^  4. 


82.  Exercise. — Prove  the  equations  of  Art.  80  directly  from  the 
figure. 

Proposition  18. 

83.  Theorem.— The  equations 

Y  =  n  fWS  P  ti  —  n  r»o«  P  —  n  air»  P 

JU          &J   OUo    ~ij  U  —  tJ   L/Uo         —  fJ  Sill 

transform  from  rectangular  to  polar  co-ordinates,  the  pole 
being  at  the  origin,  and  the  axis  of  x    the  initial  line. 


For,  yx  =  90,  and  from  Art.  62, 
=  p  cos  p ,  and  y  —  p  cos  p . 

•As  £f 


cos 


=  cos(£-Stf0Wsin?,     .'.     v=/>sinp. 

\*f/  9  3J  1/9  $ 


These  equations  are  often  written 

x  =  p  cos  6,    and  y  —  p  sin  6. 


TRANSFORMA  TION. 


35 


84.  Schol.  —  If  the  initial  line  is  not  the  axis  of  x,  we  shall  have 
(Art.  49), 

,     and  3,  =  ,  sin 


*  =  />  cos 
The  initial  line  being  x1. 

85.  Exercise.  —  Prove  that  the  equations 

x  sin  x  =  p  sin  p  ,     and  y  sin  y  =  p  sin  ^ 
y  y  xx 

transform  from  oblique  to  polar  co-ordinates. 


Proposition  19. 

86.  TJieorem.—The  equations 

transform  from  polar  to  rectangular  co-ordinates. 
From  Art.  83 

x?  =  f)2  cos2  6,     and  y2  =  pz  sin2  0 
. ' .     adding,          x*  +  yz  =  p2  (sin2  6  +  cos2  6) 

.'.    by  trig.          x2  +  y2  —  p2,     and  p  =  [/x2  +  y2 


cos  £= 

X 


also, 


COS  0=  st  = 


87.  Example. — Refer    the    point    whose    polar    co-ordinates    are 
\5,  ^ j  to  rectangular  axes  with  the  origin  at  the  pole,  and  the  axis  of 
x  for  the  initial  line.  Ans.  x  =  4-33,  y  —  2.5 

88.  Exercises. — Prove  from  the  figure  given,  that  the  equations, 

^" 

p  ,     f        pf  • 

P  x          Po~T  P      0     x 


0          p0 


0 


A 


36  THE  POINT. 


transform  from  one  system  of  polar  co-ordinates  to  another,  in  which 
0  is  the  primitive  pole,  and  0'  the  new,  p  the  old  radius  vector  and  />' 
the  new,  00'  X  the  initial  line,  and  OO  =  pQ. 

The   usual  method,  however,  is  to  transform  to  rectangular  axes, 
move  the  origin,  and  then  transform  to  polar  co-ordinates. 


Proposition  £0.f 

89.  Theorem.— The  following  are  the  equations  of  trans- 
formation, when  the  origin  is  moved  to  the  point  fa,  y0), 
at  the  same  time  that  the  directions  of  the  axes  are 
changed. 

For  by  Art.  24  the  equations 

x  =  x"  —  XQ",     and  y  —  y"—  y0/r 

will  change  the  axes  to  a  parallel  position.  On  substituting  these 
values  of  x  and  y,  and  then  omitting  the  seconds,  as  they  are  not 
needed  longer  to  distinguish  the  different  systems  of  axes,  we  have, 


1.  Tlie  equations 

(x  -  x0)  sin  xy  =  xf  sin  *'  +  y'  sin  *' 

(y  —  y0)  sin  x  =  x'  sin  Xx  +  y'  sin  yx 

transform  from  oblique  axes  to  oblique. 

2.  The  equations 

x  —  x0  =  xf  cos  %'  +  y'  cos  &.' 


transform  from  rectangular  to  oblique  axes. 


TRANSFORMATION.  37 


3.  The  equations 

...        v' 
x  —  xQ  =  x  +  y  cos 


transform  from  rectangular  to  oblique  axes  when   x    is 
parallel  to  x'. 


4-  The  equations 


x—x0  =  x  cos 


transform  from  rectangular  to  oblique  axes  when    y    is 
parallel  to  yf. 


5.  The  equations 

(x  -  XQ)  sin  xy  =  x'  sin  *'  +  y'  cos  * 

(y  -  y0)  sin  %  =  xf  sin  jjf  +  y'  cos  ^ 

transform  from  oblique  axes  to  rectangular. 

6.  The  equations 

(z-rgsm*  =  z'sin 


transform  from  oblique  axes  to  rectangular  when    x    is 
parallel  to  x'. 


38  THE  POINT. 


7.  The  equations 


(y  -  y0)  sin  *=  x'  sin  f  +  y'  cos  *' 

transform  from/  oblique  axes  to  rectangular  when   y    is 
parallel  to  yf. 


8.  The  equations 

x  —  x0  =  x'  cos  *'  —  y'  sin  % 

y-yQ  =  x'  sin  %  +  y'coa% 
transform  from  rectangular  axes  to  rectangular. 

9.  The  equations 


transform  from  oblique  to  polar,  co-ordinates. 


10.  The  equations 


x-x0=pcospx 


transform  from  rectangular  to  polar  co-ordinates. 


CHAPTER   III. 

THE  RIGHT  LINE. 

Proposition  1. 
90.  Theorem.— The  equation 

y   y\   2/2    y\ 


represents  a  right  line  through  two  given  points;  in  which 
x  and  y  are  the  co-ordinates  of  any  point  of  the  line, 
and  XL  and  y^  x2  and  y^  are  those  of  the  given  points. 


For,  let  P  be  any  point  of 
the  line  through  Pl  and  P2, 
the  given  points.  Then 
from  similar  triangles 
P«  :  P.ft  :  :  P&  :  P.& 


^ 


>>  \ 


a      \ 


\     X 


This  right  line  is  conceived  of,  as  indefinitely  extended  in  either  direction,  and 
is  called  the  locus  of  P.  It  may  be  drawn  across  any  angle,  first,  second,  third  or 
fourth,  according  to  the  position  of  PI  and  PZ. 


mi         ,.  ,,      T  ,. 

91.  Schol.  1.  —  Ine  equation  -  -  expresses  the  relation 


#  # 


that  must  hold  in  order  that  some  point  (x3,  y3),  (i.  e.,  P3),  shall  be 
upon  this  line. 


39 


40  '    EIGHT  LINE. 


For  evidently  P3  must  coincide  with  some  one  of  the  infinite  number 
of  positions  of  P.     Clear  of  fractions  and  we  have, 

yl  (x  2  —  #3)  +  y2  (xs  —  x^)  +  y3  (#1  —  #s)  =  0, 

which,  by  Art.  31,  is  the  relation  which  holds  when  a  triangle  reduces 
to  a  straight  line.  This  is  called  the  equation  of  condition  .that  three 
points  shall  be  upon  one  right  line. 

92.  The  distances  OA  and  OB  are  called  intercepts,  being  the  parts 
of  the  axes  between  the  origin  and  the  line.     It  will  be  convenient  to 
use  a  and  b  to  denote  intercepts  on  x  and  y  respectively. 

93.  Examples  in  either  rectangular  or  oblique  co-ordinates. 

(1.)  What  is  the  equation  of  the  line  through  the  points  (3,  2)  and 
(2,— 4)?  Ans.y  =  x  —  19. 

(2.)  Through  the  points  (2,—S)  and  (—4,  1)? 

Ans.  3y  =  —2x  —  5. 

(3.)  Are  the  points  (2,  5),  (1,  —  1)  and  (—1,  —9)  on  the   same 

straight  line  ? 

(4.)  The  points  (3,  4),  (1,  —  1)  and  (—3,  —  5)? 

(5.)  Write  the  equations  of  the  lines  through  the  points  in  ex.  (3), 
and  in  ex.  (4). 

(6.)  Draw  the  lines  whose  equations  are  obtained  in  the  examples  of 
this  article. 

94.  Exercise. — Show  that  the  form  of  the  equation 

y  —  y\  _  3/2  —  y\ 

X  —  Xi        X2  —  Xi 

is  not  changed  by  moving  the  origin  to  any  point  (a:0,  y0). 


Proposition  2. 
95.  Theorem.— The  equation 

-+H 

a     b 
represents  a  right  line ;  in  which  x    and   y    are  the  co-ordi- 


GENERAL  EQUATION  IN  TERMS  OF  INTERCEPTS.          41 

nates  of  any  point  of  the  line,  and    a    and    b    are  the  inter- 
cepts. 

For,  in  Art.  90  let  Pl  Ml 
on  the  axis  of  x,  and  P2  on 
the  axis  of  y. 


.  • .     xl-—  OPl  =  a,  and  y^  —  0 

x2  —  0,  and  y2  =  OP2  =  b 

y—0     b-0 
.'.     we  have     - — 

x  —  a     0  —  a 

x_+l=L         /  \ 

a     b 

This  equation  is  symmetrical,  and  of  the  zero  degree  when  we  consider  both 
x  and  y  and  also  the  constants  a  and  6.  In  x  and  y  only,  it  is  of  the  first 
degree. 

96.  When  a  and  b  are  given,  we  have  the  equation  of  a  par- 
ticular line.     But  a  and  b  may  represent  any  intercepts,  and  in 
this  sense  the  equation  is  said  to  be  the  general  equation  of  a 
right  line  in  terms  of  its  intercepts. 

97.  Examples. — Reduce  the  equations  of  the  straight  lines  obtained 
in  Art.  93  to  the  form 


a^b' 

i.  e.,  so  that  the  right  hand  member  is  +1: 
e.g.,    if  8y  =  —  x  —  7 

then,  transposing  and  dividing, 


x 


y~=i 


. ' .    the  intercepts  are  — 7,  and  :z^. 
Also  show  that  when  (Art.  90) 

Knv.  =  Ji=£  then  a=£!^Mii  and  6  =  ,= 


42 


RIGHT  LINE. 


Proposition  3. 

98.  Theorem.—  The  equation 


= 


—  X1 


represents  a  right  line  through  one  given  point ;  in  which 
x    and    y    are  the   co-ordinates   of  any  point  of  the  line, 


sin 


Xi    and   T/!    those  of  the  given  point,  anci  m  = . 


sin 


For,  from  the  triangle  PQPl  by  trig. 

y  —  yl :  x  —  xl :  :  sin  QP^P  :  sin  P-^PQ, 
.     i 

y—y^      n  x 

=  m, 


x  — x 


i      sin 


y 


in  which  I  denotes  the  direction  of 
the  line — i.  e.,  sin  x  is  read,  "  sine  of 
the  angle  between  the  axis  of  x  and  the  line  V 


10 


sin  , 

99.  Cor.  1.— When  yr  =  <*  =  £0°,  -  then  m =  tan- 

JC  I  JC  ' 

cos  x 

.'.     in  rectangulars  m  =  the  tangent  of  the  angle  which  the  line  makes 
with  the  axes  of  x. 


100.  Cor.  *.- 


we  have, 


sin 


x  — 


COS 


x 

—  xl 


sn 


cos 


in  which  I  =  PPl  is  the  distance  of  any  point  P  from  the  given  point 
Plt  and  is  measured  along  the  line. 

This  equation  may  also  be  written 


EIGHT  LINE   THROUGH  ONE  GIVEN  POINT.  43 

101.  Schol.  1.  —  It  is  to  be  noticed  that  when  the  value  of  m  is 
given  in  the  equation 


a  particular  line  is  represented  ;  when  it  is  not  given,  the  equation  may 
represent  any  one  of  the  lines  through  P1;  and  in  this  sense  it  is  said 
to  be  the  general  equation  of  the  line  through  one  given  point. 

102.  Schol.  2.  —  In  Art.  90  suppose  that  P2  coincides  with  Plt 
then,        -  -  1  —  —  =  m  =  some  indeterminate  quantity. 

This  algebraic  indetermination  expresses  the  known  fact  that  an  infinite 
number  of'  different  lines  can  be  drawn  through  PI. 


Proposition  4. 
103.  TJieorem.—The  equation 

y  —  b  =  mx 

represents  a  right  line;   in  which   x    a?id   y    are  the  co- 
ordinates of  any  point  of  the  line,    b    is  the  intercept  on  y 

sin^ 

and  m= - 

sin  ^ 

For,  in  Art.  98  let  Pl  fall  upon  the  axis  of  y. 
xl  =  0    and  yl  =  b, 

y-b 
.  • . =m,     or  y  —  b  =  mx. 

Similarly  if  Pl  be  upon  the  axis  of  x, 
then          xl  —  a,     and  yl  =  0. 


y  y        / 

— - —  —  m.     or  x  —  a  =  —. 
x  —  a  m 


44  RIGHT  LINE. 


104.  Cor.  1. — When2/  =  £0°,  then  m  =  tan  l    or  — ==cot* 

x  m 

The  equation  y  —  b  =  7712:    might  therefore  be  written, 
y  —  b  =  x  tan  ^ ,     or  y  =  #  tan  *  4-  ft. 

105.  Cor.  £. — If  5  =  0,   y  =  ma;   is  the   equation  of  a  right   line 
through  the  origin — i.  e.,  if  the  equation  contains  no  constant  term,  then 
the  origin  is  on  the  line. 

If  y  =  x,  the  line  bisects  the  angle  ^,  and  passes  through  the  origin, 
and  y  =  x  +  b  is  parallel  to  this  bisecting  line. 

106.  Cor.  3. — When  m  =  0,  then  y  =  5,  and  the  line  is  parallel  to 
the  axes  of  x. 

When  also   5  =  0,  then  y  =  0,  and  the  line  coincides  with  the  axis 
of  #. 

When  in  the  last  equation  of  Art.  103,  ~  =  0,  then  x  =  a,   and  the 

line  is  parallel  to  the  axis  of  y. 

When  also   a  =  0,   then  x  =  0,  and  the  line  coincides  with  the  axis 
of  y. 

107.  Examples. — (1.)  Determine  the  intercepts  of  the  right  line 
through  the  point  (1,  5),  when  m  =  - .  Ans.  a  =  —  5-| ,  5  =  4\  - 

(2.)  Determine  the  intercepts  of  the  right  line  through  the  point 
(4,  —  <£),  when  m  =  f .  ^.ws.  a  =  7^ ,  b  =  —  -?^. 

(3.)  Find  the  equation  and  intercepts  of  a  right  line  through  the 
point   ( — 2,  — 4),  and  perpendicular  to  the  axis  of  y,  when  at  =  ^. 
-4ns.  Equation  is  y  =  —  |  —  5.     Intercepts,  a  =  —  10,  b  =  —5. 

(4.)  Find  the  values  of  m  in  the  examples  of  Article  93. 

(5.)  Show  that  the  intercept  a  = . 

m 

108.  Exercise. — Derive  the  equation  of  Art.  103  from  the  figure, 
and  prove  the  equations  of  Arts.  90  and  98  from  it,  when  at  =  90°. 


THE  GENERAL  EQUATION  OF  THE  RIGHT  LINE.          45 

Proposition  5. 
109.  Theorem.—  Tlxe  general  equation  of  the  first  degree 


represents  some  right  line;  in  which  x  and  y  are  the 
co-ordinates  of  any  point  of  the  line,  and  A,  B  and  C 
may  each  have  any  real  value  whatever. 

For  if  the  origin  be  moved  Fv     \-^ 

to  a  parallel  position  by  the 
equations  of  (Art.  23),  viz. : 
x=x0+x',  and  y=yQ  +  y', 
we  obtain  by  substitution, 


,  ,  p^ 

0  .  .  .  (a.)        \ 

Values  for  x0  and  y0  have  not  yet  been  assigned,  and  we  may 
evidently  give  them  whatever  values  we  please.  Let  them  have 
such  values  that  AxQ  +  ByQ+  C=0.  .  .  .  (&.) 

There  can  be  an  infinite  number  of  such  values  —  i.  e.,  of  positions 
of  the  new  origin  (XQ,  y0),  —  for  if  A,  B  and  C  are  given,  and  we 
assign  any  value  whatever  to  y0,  we  can  evidently  find  from  eq.  (6.) 
a  corresponding  value  of  x0  such  as  will  verify  the  equation.  Let 
the  new  origin  (xQt  y0)  be  at  any  point  0',  that  satisfies  eq.  (6.)  ; 
then  eq.  (a.)  becomes  Ax'  +  By'  =  0.  ..........  (c.) 

Equation  (c.)  then  represents  the  same  thing  referred  to  new 
axes  that  (a.)  represented  when  referred  to  the  primitive  axes. 

If  in  (e.)  x'  =  0,  then  y'=0,  which  shows  that  this  new 
origin  is  a  point  on  the  line,  straight  or  curved,  which  is  repre- 
sented by  (a.)  and  (c.). 

We  may  omit  the  primes  and  write  eq.  (c.),  Ax  +  By  =  0, 
for  the  primes  are  used  only  to  distinguish  conveniently  one 
system  of  axes  from  another. 

Now  multiply  by  sin  ^, 


or  (Art.  13),        Ax  sin  *  -  By  sin  yx  =  0. 


46  RIGHT  LINE. 


Next  change  the  direction  of  the  axes.    Substituting  from  Art.  69, 
,-.    A(x'  sin  xy  +y'  sin*')  -B  (xf  sin  %  +y'  sin  £)  =0. 

Rearranging  the  terms  we  obtain, 

(A  sin  *'-.5smf)  xf  +  (A  sin  &  -B  sin  %)  y'  =  0.  .  .  (d.) 

Since  the  angles  *',  ^,  ^,  *  ,  are  not  yet  determined,  and  two 
of  them,  either  *'  and  *',  or  *'  and  g',  or  *  and  *'  or  %'  and  *'> 
are  independent,  we  may  assign  any  values  we  please  to  the  two, 
or  affix  such  conditions  as  will  determine  their  values.  Let  the 
two  following  conditions  hold  : 

1st.  A  sin  y'  —  B  sin  *'  =  0,     and  2d.  A  sin  *'  -  B  sin  |'  <  0. 
Substituting  the  1st  condition  in  equation  (d.),  we  have, 


by  the  2d  condition.  From  which  we  see  that  there  is  no  point 
represented  by  this  equation  which  does  not  coincide  with  the 
axis  of  xf,  for  yf  =  0  is  evidently  the  equation  of  a  line  coinciding 
with  the  axis  of  x',  but  the  axis  of  xf  is  a  right  line.  .  •  .  (a.)  re- 
presents a  right  line. 

110.  Cor.  1.  —  To  reduce  the  general  form  of  the  equation  of  a  right 
line, 

Ax  +  By  +  C=  0, 

to  the  form  of  the  equation  in  terms  of  the  intercepts  (Art.  95),  trans- 
pose and  divide  by  —  Q. 


,     or, 


,        , 


is  the  form ;  in  which 

_C_ 
A  ' 
are  the  intercepts. 


ONE  LINE  IN  TERMS  OF  TWO  OTHERS.  47 

111.  Cor.  2. — To  reduce  the  general  form  to  the  form  of  one  inter- 
cept (Art.  103),  solve  with  reference  to  y, 

A         C 


C                A      s*n  x 
in  which,  as  before,         b  = — ,    and  —  —  = 

y 

112.  Schol.  1. — The  equation 


represents  two  lines  separately,  for  by  the  general  theory  of  equations 
each  factor  =  0. 


113.  Schol.  2. — If  Al  x  +  B\  y  +  Ci  =  0  and  A.z  x  +  •< 

are  the  equations  of  the  lines  (1)  and  (2),  then  A^x^-\-  Blyi-\-  Ci  =  0 
and  Az  Xi~\-  -52  2/t  +  Cz  =  0  hold  respecting  the  co-ordinates  of  the 
point  of  intersection,  (xh  yt).  Eliminating,  we  have, 

C\  B>i  —  C2  -JDi          ,  C/i  AZ      C 2  -A-\ 


114.  Schol.  3,  —  The  general  equation  of  the  first  degree,  viz., 


contains  but  two  arbitrary  constants. 

JV.  B.  —  We  shall,  for  convenience,  speak  of  "  the  line  Ax+By+C=  0" 
meaning  the  line  which  the  equation  Ax  +  By  +  0=  0  represents. 


Proposition  6. 
115.  Theorem.— The  equation 

k,  (A,x  +  By  +  Cl)  +k2 

represents  some  right  line  passing  through  the  intersection 
of  the  lines  Alx  +  Bly+Cl  =  0  and  A&  +  By  +  d  =  0 ;  in 
which  Aa  and  h  are  any  multipliers. 


48  RIGHT  LINE. 


For,  by  Art.  109  it  is  the  equation  of  some  right  line,  since  it 
may  be  reduced  to  the  form, 

(Afa  +  Ajc2)  x  +  (Bfa  +  BJc2]  y  +  Cfa  +  CJc2  =  0, 
or,  as  it  may  be  written,     A3x  +  B$  +  C3  =  0. 

Moreover,  the  equation  is  evidently  satisfied  when  both 
Ap  +  By  +  C^O,   and   A&  +  B#+C2  =  0, 

provided  the  values  of  x  and  y  are  the  same  —  i.  e.,  simultaneous, 
in  the  two  expressions.  But  x  and  y  can  have  the  same  values  in 
lines  (1)  and.  (2)  only  at  their  intersection  ;  therefore  the  line  (3) 
passes  through  the  intersection  of  lines  (1)  and  (2). 

116.  Cor.  1.  —  The  equation  of  line  (1)  is,  —  k  (A&  +  By  -f  (?)  =  0. 

The  equation  of  line  (2)  is,  —  &2  (A.2x  +  By  +  (?2)  =  0,  and  that  of  line 
(3)  is,  h  (A&  +  By  +  a)  +  £2  (A&  +  By  +  Q  =  0.  Add  these 
together,  and  they  vanish  identically.  Therefore,  when  the  equations 
of  three  lines,  after  being  each  multiplied  through  by  any  constants 
&lf  £2  and  £3,  can  be  added  so  as  to  vanish  identically,  the  lines  pass 
through  one  common  point,  and  are  called  convergents,  or  a  pencil  of 
three  rays. 

117.  Cor.  2.  —  If  lines  (1),  (2)  and  (3),  whose  equations  are, 


intersect  in  a  point,  by  elimination  we  obtain, 
,      G  (A  A  ~  A,B2)  +  a  (A3B,  -  A,B,)  +  a  (A,B,  -  A,B,}  =  0, 
which  is  the  equation  of  condition  that  three  lines  intersect  in  a  point. 

118.  ScJiol.  —  When  the  constants  of  lines  (1)  and  (2)  have  definite 
numerical  values,  then  (1)  and  (2)  are  determinate  lines.  But  in  com- 
bining (1)  and  (2)  to  obtain  (3)  (Art.  115),  it  is  still  possible  to  assign 
at  pleasure  any  values  to  ^  and  #2,  thus  producing  any  one  of  an 
infinite  number  of  lines  (3),  each  of  which  passes  through  the  inter- 
section of  (1)  and  (2),  and  each  of  which  satisfies  the  equation  of  con- 
dition in  Art.  117. 

E.  G.,  if  line  (1)  is  x  +  y  +  2  =  0,  and  line  (2)  is  x  —  2y—l  =  0, 
then  line  (3)  is,    2x  —  y  +  l  =  0,    when    £t  :  &2  =  1, 
and  is,    3x  +  3  =  0,    when  ^  :k2  =  %,    etc.,  etc. 


RELATION  OF  THREE  ANGLES  OF  A    TRIANGLE.         49 

119.  Exercise.  —  Prove  that  the  three  lines  (1),  (2)  and  (3)  enclose 
the  triangle  whose  area  is  t,  when 

[a  (A&  -  A,S,)  +  a  (A*B,  -  A  jg.)  +  a  (A  A  -  A, 

(A  A  -  4A)  (A  A  -  A&)  (A&  -  AA) 

CJA-C 

by  subshtutmg        *,  =          . 


V,  = 


etc., 


in  the  formula  of  Art.  31.     Also  prove  Art.  117  from  the  above. 

Proposition  7. 
120.  Tlieorem.—The  equation  ^Y 

tan  £  tan  £  tan  [» =  tan  Jj  +  tan  'j  +  tan  *J  ' 

*/ 
expresses  the  relations  which  subsist  between  the  angles? 

any  plane  triangle  in  which    IQ.    /x    and    /2    are  the  direc 
tions  of  its  sides.    (Art.  12.) 

Y 

For  (Art.  12),  360°  =  \  +  \\  +  ^ 

this  being  the  sum  of  the  external 
angles  of  the  triangle. 

>_fc_Jl  _1_  ^0 


-</ 


a,  o 


,  x      tan   1  +  tan  ? 
and  by  trig.  tan    (360°-  2)  =  -  - 

17 


=  -tan 


7 


-  tan  g  ton 


Cl^ar  of  fractions,   . ' .    tan  £  tan  £  tan  Jj  -  tan  £  +  tan  £+  tan 


Proposition  8. 
121.  Theorem.—  The  equation 


expresses  the  value  of  wio, 
any  two  lines    y  =  m^x  H-  61, 


tangent  of  the  angle  between 

2/  =  w^  +  6«. 


50  EIGHT  LINE. 


For,  let  the  axis  of  x,  —  i.  e.,  the  line  y—0^  having  the  direction 
lo,  together  with  the  line  y  =  m^  +  6D  having  the  direction  llt 
and  the  line  y  =  m^o  +  b2,  having  the  direction  Z2,  form  a  triangle, 

then  (Art.  120),  ra0  =  tan  ^,    ml  =tan  *!,    m2=tan  *?. 

l       tan  £  +  tan  * 

By  Art.  120,       -  tan  £  = 

^ 


.-.     (Art.  13) 


1  -tan  1  tan  f 

.C  t2 

tan     -  tan 


0.    .,    ,  mt  —  m0          ,-. 

Similarly,    m2  =  —  -  -  .  .  .  (/•) 


Cor.  l.—U     =  0,     then     <m»  =  0. 
.  •  .     (e.)  reduces  to    ml  —  m.2  =  0. 

But  (Art.  Ill)  m,  =  41,  and  m,  -  ^ 


Hence  the  condition  of  parallelism  between  the  two  lines  is, 
m^—m^O,   or—  '  —  —  ^  -  =  0, 

X>!         X)2 

as  may  also  be  seen  from  Art.  113. 

123.  Cor.  2.—  If  £  =  00°,   then  m0  =  oo. 

.  *  .     (e.)  reduces  to   1  +  ^777-2  =  #,   or  mx  =  --  , 


.  •  .     (Art.  Ill)         A,A2  +  B^  =  0. 
Either  equation  is  the  condition  of  perpendicularity  of  two  lines. 

124.  Cor.  3.—lil^  =  01   then  m,  =  0,  or  (Art.  Ill)      —  2  =  0', 


PERPENDICULARITY  AND  PARALLELISM.  51 

.*.     A2  =  0,   is  the  condition  of  parallelism  to  axis  of  x,  —  i.e.,   the 
line   B$  +  02  =  0  is  parallel  to  the  axis  of  x. 


If  b  =  90°,  then   w2=oo,   or  (Art,  111)   —     =  00; 
x  -B* 

.*.    £  =0  is  the  condition  of  perpendicularity  to  the  axis  of  x,  — 
i.  e.,  the  line   A^c  +  (72  =  0  is  perpendicular  to  the  axis  of  #. 


125.  Cor.  4.-Since  tan     =-  =         ll,  (Art.  Ill), 

.        .         .      ft  _ 

'    ^  "" 


m?)  ^ 
and  cos       =-- 


126.  Examples.  —  (1.)  Given  the  equation 


to  construct  the  three  intersecting  lines  indicated  by  the  equation,  and 
find  the  co-ordinates  of  their  point  of  intersection. 


(2.)  Show  numerically,  by  means  of  Art.  121  and  the  equations 

y  =  3x  +  5,     4y  =  x  +  8     and    y  =  —  x  —  l, 
that  the  relation  of  Art.  120  holds. 

127.  Exercise.  —  Prove  that  when  the  axes  are  oblique, 


Proposition  9. 

128.  Tlieorem.— (Rectangular  co-ordinates^) 
^e  line  y  =  mtf  +  62  is  parallel  to  y  =  m^  +  bi  by  Art. 
The  line   y  —  yl=m1(x  —  xj    passes  through   (a*,  7/1),   and  is 
also  parallel  to    y  =  mtf  +  62. 


52  RIGHT  LINE. 


The  line  A&  +  B$  +  C2  =  0  is  parallel  to  A&  +  B^y  +  Ci  =  0 
by  Art.  1%%. 

The  line  Al  (x  —  xj  +  Bi(y  —  y^  =  0  passes  through  (xlt  yj, 
and  is  parallel  to  A&  +  B$  +  Ci  =  0. 


The  line     y  •=  --  x  +  b-2     is  perpendicular  to     y  =  m&  +  bv 

by  Art.  1%3.          *"* 

_  2 
The  line    y  —  y\  =  -   (»  —  «i)    is  perpendicular  to    y=  m& 

m: 
4-  b    and  passes  through     (xi,  2/1).      The  same  line  is   also 

perpendicular  to    y  —  yl  =  ml  (x—x^)    at  the  point    (x1?  7/1). 

The  line    B&  —A^y  +  C2  =  0    is  perpendicular  to    A&  +  B^y 
C,  =  0    by  Art.  123. 

The  line  Bl  (x  —  x^  —  Ai(y  —  2/1)  =  0  is  perpendicular  to 
AiX  +  Biy  +  C!  =  0,  and  passes  through  (xi,  2/1).  It  is  also  per- 
pendicular to  A!  (x  —  Xi)  +  Bi  (y  —  2/1)  =  0,  at  the  point  fa,  yj. 

129.  Example.  —  Find  the  equations  of  two  lines  at  right  angles 
with  each  other,  the  first  of  which  passes  through  the  point  (—2,  5), 
and  has  tan  |==#i  while  the  second  passes  through  the  point  (4,  1). 

Ans.     =  2x^9  and  2    +  x  =  6. 


130.  Exercises.  —  (Oblique  co-ordinates^)     Show  that 


_  , 

—  x  +  ia,  and  y  =  m^x  -f  ^  are  perpendicular. 
ml  +  cos  y 


X 


(2.)  (^cos^-^O^-CA 

is  perpendicular  to  A&  -f-  B^y  +  ,-Q  =  0. 

ljrrnl  cos  y 


COS  * 
# 


passes  through  (#1,  ?/i),  and  is  perpendicular  to    y  =  ml  x  +  &i- 

(4.)         (vl,  cos  y  -  £,)  (a-  -  *0  -  (B,  cos  g  -  A,)  (y  -  y,)  =  0 
passes  through  (a;lt  i/J,  and  is  perpendicular  to  A&  +  B$  +  Cl  =  0. 


ONE  LINE  AT  A   GIVEN  ANGLE  TO  ANOTHER.  53 

Proposition  JO.f 

131.  TJteorem.—  (Rectangular  co-ordinates.}     The  equations 

y  =  m2x  +  b2,      and    y  —  yi  =  'm2  (x~  xz), 
which  may  by  Art.  121  be  written 

ml  —  mQ  ml—m0 

y=T  ,  --  X  +  OB       and     y  —  y=-—^         -(x  —  x\ 
1  +  ra^o  1  +  m^  v 

represent  lines  making  an  angle  with  any  given  line 

y=mlx  +  b1,     or    y  —  yl  =ml  (x  —  arj, 
such  that  the  tangent  of  this  angle  is   m0  =  tan  ^. 

This  is  evident  from  Articles  103  and  121. 

It  is  also  to  be  noticed  that  two  lines  can  be  drawn  each  making 
an  angle  of  the  same  number  of  degrees  with  y  =  mrr  +  61;  but 
one  makes  a  positive  and  the  other  a  negative  angle,  correspond- 
ing to  +  m0  and  —  m0  respectively,  from  which  we  have  two  values 
of  ra2. 

132.  ScJiol.  1.  —  If  mi  =  0,  the  given  line  is  parallel  to  the  axis  of  ar, 
.  •  .     y  =  zp  m^c  +  &2    and    y  —  y\  =  T  m0  (x  —  x^)     are  the  equations. 


133.  Schol.  2.  —  If  mi  =  00,  the  given  line  is  perpendicular  to  the 
axis  of  a:.     Since  in  this  case 


rax 


1  -J- 

.  •  .     y=±  —  x  +  Z»2,     and    y  —  yl  =  ±  —  (a?  —  Xi) 
m0  mQ 

are  the  equations. 

134.  Schol.  3.  —  If  m}=m0,  then   y  =  b.2  and  y  —  ^  =  0  are  the 

equations.     But  if  m^  =  —  mo  the  equations  become 


or  =  a:an  2,  and  y  —  y,=  (a;-  a;,)  tan  ^  ('|). 


54  RIGHT  LINE. 


135.  Schol.  4. — If  mi  =  —  — ,  then  x  =  0  and  x  —  xl  —  0  are  the 

equations.     But  if  ml  =  — ,  then  the  equations  become 

y  —  x  cot  2  I  £  )  +  b.z  and  y  —  yl  —  (x  —  x^)  cot  2  I  £  J. 

136.  Schol.  5. — If  m0  =  0,  the  line  is  parallel  to  the  given  line,  and 
the  equations  become    y  =  m^x  +  #2.  and  y—y\'=rin^(x  —  a^). 


137.  Sc/ioJ.  tf.-If  m0  =  oo,  then  J^L^L  =  !^ =  _  J_t 

1  +  WiW0        ^  m 

+  W! 

m0 

.  • .    y  = x  -\-  b2,  and  y  —  y^  = (x  —  x^\  are  the  equations, 

ml  m-i 

and  the  line  is  perpendicular  to  the  given  line. 

138.  Examples. — (1.)   Form  the   equation  of  a  line   making  an 
angle  of  30°  with  the  line  7y  —  x-^/3  +  2  =  0,  and  having  an  intercept 
on  y  of  — 4- 

Am.  y=  —  —7 -/S.x—4,  or    y  =  j^v/3,x-4. 

(2.)  Form  the  equations  of  two  lines  making  with  each  ether  an 
angle  of  -£5°,  the  first  passing  through  the  two  points  (1, 2)  and  (— 4>  —  #), 
and  the  second  through  the  point  (1,  —3). 

Ans.  y  =  x-\-l,  and  y  =  — 3,  or  x  =  1. 


Proposition  11. 
139.  Theorem.— The  equation 

x  cos  ^  +  y  cos  -^  —p  =  0 

represents  a  right,  line;  in  which  p  is  the  length  of  the 
perpendicular  let  fall  from  the  origin  upon  the  line,  and 
P.  and  P  are  the  angles  between  the  co-ordinate  axes  and 
the  perpendicular. 


EQUATION  OF  LINE  IN  TERMS  OF  DIRECTION  COSINES.  55 
Let  ^  + 1  —  1  =  0  represent  AB  (Art.  95).     Multiply  by  p, 


f)        f) 
*'•     ax  +  b 


By  trig.  -  =  cos  ^,  and  -^  =  cos  Jf 

.  * .  x  cos  -^  +  y  cos  P  —p  —  0. 

This  is  the  equation  of  a  right  line  in  terms  of  the  direction 
cosines  of  its  perpendicular,  and^>  is  always  considered  as  positive. 

The  equation  may  also  be  derived  directly  from  the  figure, 
Oe  +  cd  =  Oc.  cos  eOc  +  cP.  cos  dcP  —  y  cos  y  +  x  cos  *  =p. 

140.  Cfcr.J.— If  ?  =  00°, 


y 


P  =  ^  —  y  =-P  —  90* 

x       xxx 


X 


a;  cos  |!  +  y  sin  P  —p  =  0, 

or,     if  a  —  P,     #  cos  a  +  y  sin  a  —  £>  =  0. 


141.  Cor.  £.—  The  value  of  p,  from  Art.  139,  is^  =  a  cos  ?  =  a  sin  J.- 

But  by  trig,  we  have  from  the  triangle  AOB 


sm;  = 


ab  sin 


56  RIGHT  LINE. 


142.  Cor.  3.— To  reduce  the  form  Ax  +  By  +  0=0  to  the  form 
x  cos  -^  +  y  sin  -^  — ^?  =  0,  let  ^  =  #0°  in  the  formula  of  the  preceding 

article;  then p  =  —        — — -.     By  Art.  110,     a  =  —  -7,     6  =  —  -- 
y  (or  +  o )  -a,  .o 


c 


. ' .     the  form  is x  4- y  =  — 


Hence  to  perform  the  reduction  in  any  case,  divide  through  by 
I/'' (A'2  +  j52).  It  is  to  be  noticed  that  if  the  reduction  be  applied  to 
itself,  the  divisor  is  of  the  form  j/(sin2  a  -f  cos2  a)  =  1,  which  does  not 
change  the  form. 

143.  Example. — Form   the    equations,    in   terms   of   the    perpen- 
dicular from  the  origin  and  its  direction  cosines,  of  the  diagonals  of  a 
parallelogram,  each  of  whose  sides  is  4,  and  one  of  whose  angles  is  60*, 
taking  two  adjacent  sides  as  axes  of  reference 

Ans.  ~2  xyll  +  ~^y  y^  —  8.464  =  0,  or  x  +  y  =  4, 

and  x  cos  60°  —  y  cos  60°  =  0,     or  x  =  y. 

144.  Exercise. — Prove   that  with  oblique  axes  the  divisor  corre- 
sponding to  that  in  Art.  142  is 


Also,  show  that  for  the  line    y  —  b  =  mx    the  divisor  becomes 
!/(-?  +  2m  cos  ID  4-  m2) 


sin  to 


PERPENDICULAR  FROM  A  POINT  TO  A   LINE.  57 

Proposition  12.-\ 
145.  Tfieorem.—The  equation 

±  (xl  cos  a  +  yl  cos  ft  -p)  =pl 

expresses  the  length  of  a  perpendicular  let  fall  from  any 
point  upon  a  given  line;  in  which  (#1,  y^  is  the  point,  pv 
is  the  length  of  the  perpendicular,  and  x  cos  a  +  y  cos  /5  —  p  =  0 
is  the  equation  of  the  given  line. 

For,  if  x  cos  a+  y  cos  ft  —p=  0 

is  (Art.  139)  the  equation  of  /     Pj.,^^  \  _  XL 

AB  referred  to  0,  and  we  move 
the  origin  to  P^  (Art.  23),  the 
equation  of  AB  then  becomes 


#!  cos  <JL  +  yl  cos  ft  +  xr  cos  a  +  yf  cos  ft  —p  —  0. 

But  xr  cos  a+yr  cos  ft  —pl  =  0  is  the  equation  of  AB  referred 
to  Pr  Substituting,  we  obtain  —  (xl  cos  a  +  yl  cos  ft  —p)  —pv. 
"When  Pl  is  on  the  side  of  AB  opposite  to  the  origin,  evidently  the 
perpendicular  pl  is  negative,  since  it  is  let  fall  in  a  direction 
opposite  to  that  of  p, 

.'.      ±  (xl  cos  a  +  yl  cos  ft  —p)  =pr 

146.  Cor.  —  By  a  reduction  like  that  of  Art.  142  it  can  be  shown  that 

=  Ax,  +  By,  +  C 

~ 


147.  Examples.  —  (1.)  Find  the  length  of  the  perpendicular  from 
the  point  (7,  2)  on  the  right  line,  4x  +  3y  =  10.  Ans.  pl  =  4.8 

(2.)  Find  the  '  lengths  of  the  perpendiculars  from  the  point  (  —4,  3) 
on  the  sides  of  the  triangle  whose  vertices  are  (1,  5),  (5,  1)  and  (  —  1,  —  1). 

148.  Exercise.  —  Prove  that  the  equation 

Axl  +  By,  +  C 
A  cos  0  +  B  sin  0 

expresses  the  length  of  line  drawn  in  a  given  direction  from  a  given 
point  to  meet  a  given  line  ;  in  which  r:  is  the  length,  Ax  +  By  +  O=0 
the  given  line,  0  its  inclination  to  the  axis  of  x,  and  (xlt  y^)  the  given 
point. 


58 


EIGHT  LINE. 


We  may  also  write 


—  p 


/  7 

cos    cos  0  -\-  cos    sin  0 
x  y 

E.  G.  The  length  of  a  line 
drawn  from  the  point  ( —  4>  10) 
to  meet  the  line  x  —  %y  —  3,  and 
making  an  angle  of  4$°  with  the 
axis  of  x,  will  be  found  to  be 
TI  =  — 88.05. 


\ 


A 


Proposition  13. 
149.  Tlieorem.—The  equation 

(x  cos  ^  +  y  cos  &  —  pj  ±  (x  cos  ^2  +  y  cos  ^2  —  ^>2)  =  0 

is  that  of  the  line  bisecting  the  angle  between  two  given 
lines  ;  in  which 

x  cos  ^  +  y  cos  *y  —pl  =  0,     and  x  cos  p^  +  y  cos  ^2  —  p2  =  0 
are  the  given  lines. 

For,  by  Art.  115  it  is  the  equation  of  some  right  line  as  PPZ 
through  the  intersection  of  two 
others,  as  Mm  and  Nn.     Take 
any  point  (x,  y)  on  the  line  (P, 
in  the  figure)  ;  then  (Art.  145) 


X   COS 

and 
x  cos 


cos       —     = 


cos      —— 


But  by  hypothesis  the  sum 
of  the  first  members  of  these 
equations  =  0 ;  .  • .  pn  —pm  =  0,  or  pn  =pm. 


\ 


P  is  any 
point  on  the  line  of  equal  perpendiculars — i.  e.,  upon  the  bisector. 

.  %     (x  cos  pxl  +y  cos^1-^)  +  (x  cosf2  +  y  cos^2-p2)  =0 

is  the  equation  of  the  external  bisector — i.  e.,  the  bisector  of  the 
angle  not  containing  the  origin,  as  BP2.     Similarly, 

(x  cos  %  +  y  cos  ^  — jpj  -  (a  cos  ^2  +  y  cos  ^2  -_p2)  =  0 
is  the  equation  of  the  internal  bisector  AP2. 


INTERNAL  AND   EXTERNAL  BISECTORS.  59 

150.  Cor.  1. — The  equation  in  rectangular  co-ordinates  of  the  bisector 
of  the  lines 

+  C2  =  0,  is  (Art.  142) 
By+C* 


151.  Cor.  2. — The  co-ordinates  of  the  point  of  intersection  of  the 
lines    x  cos  av  +  y  sin  a}—pl  =  0,     and    x  cos  «2  +  y  sin  «2  — p2  =  0, 

»i  sin  a.2  — p2  sin  at  ??,  cos  a.t  — p2  cos  av 

are  (Art.  113),  xi  =  —. — T^^L-( \  and  yt  = : — ; — * — r . 

sm  (a!  —  a2)  sin  («!  —  a2) 

152.  Cor.  3—  The  lines 

x  cos  «i  +  y  sin  c^  — pl  =  0, 
x  cos  «2  +  y  sin  «2  — ^?2  =  0, 
x  cos  «3  +  y  sin  a3  — j?3  =  0, 
(Art.  117),  intersect  in  a  point,  when 

^?!  sin  («2  —  aa)  +^2  sin  («3  —  aj)  +^3  sin  («!  —  a2)  =  0. 


153.  Cor.  4.—  By  Art.  118  we  have  also, 

fa  sin  (a2  —  a8)  +jpa  sin  (a3  —  aQ  +^3  sin  («!  —  «2)]2 

-  -  -  :  -  -  --  -  -         -  —    -   ~T~  &(,. 

sin  (a2  —  «3)  sin  («3  —  aj  sin  («i  —  «2)  x 

154.  Example.  —  Form  the  equations  —  in  rectangular  co-ordinates  — 
of  the  internal  bisectors  of  the  angles  of  a  triangle,  the  co-ordinates  of 
whose  vertices  are  (1,  2),  (—3,  5),  and  (—1,  —4}  I  and  show  that  they 
intersect  in  a  common  point. 

Ans.  y  =  0.312  x  +  1.692,    y  =  -1.55x±  0.35,    y  =  19.39  x  +  15.39 
Co-ordinates  of  point  of  intersection. 

x  —  —  0.72,  nearly,  y  =   1.466,  nearly. 

155.  Exercise.  —  Prove  that  in  oblique  co-ordinates 

=Q 


,  cos 

is  the  equation  of  the  bisectors  of  the  lines 
Cl  =  0,    and 


60  EIGHT  LINE. 


POLAR    CO-ORDINATES. 

Proposition  14. 
156.  Theorem.—  The  equation 


represents  a  right  line;  in  which  p  is  the  radius  vector 
of  any  point  of  the  line,  and  p  the  length  of  the  per- 
pendicular from  the  pole  upon  the  line. 

For,  by  trigonometry, 
PCOB'=J,.    But  £=£ 

•••      cos     ~    = 


„,.        ,   ,      . 

This  may  also  be  written  \ 

p  cos  (6  —  a)  —p,   or   jO  cos  a  cos  d  -\-  p  sin  a  sin  6  =p. 

157.  Cor.  J.  —  If  a  =  0,  then  />  cos  0  =p  is  the  equation  -of  a  line 
perpendicular  to  the  initial  line. 

158.  Cor.  2.  —  The  equation   0  —  c  represents  a  line  through  the 
origin  making  the  angle  c  with  the  initial  line. 

159.  Cor.  3.  —  The  angle  between  the  lines  p  cos  (0  —  a1)=p1  and 
p  cos  (0  —  «2)  =p2  is  the  same  as  that  between^  and  p2  —  i.  e., 


160.  Cor.  4.  —  Two    lines    are    perpendicular    to    each    other    if 
«2  -~  «i  =  90°,  and  parallel  if  «2  —  ax  =  0. 

161.  Cor.  5.  —  The  equation  of  the  external  bisector  between  the 
lines  (1)  and  (2)  is  (Arts.  83  and  149),  when  a>  =  90°, 

p  [cos  (0  —  a,)  +  cos  (0  -  «2)J  =pz  +plt 

-n     .  .  /m  +  n\        I'm  —  n\ 

By  trig.  cos  m-\-  cos  n  =  #cos  I  —  -  —  1  cos  I  —  -  —  I. 


POLAR  EQUATION  OF  RIGHT  LINE.  61 


Similarly  for  the  internal  bisector, 
cos 


162.  Schol.  1.  —  The   equation     p  cos  (0  —  a)=pl  cos  (Ol  —  a)     is 

the  polar  equation  of  a  line  through  the  point  (plt  0^,  for,  (Art.  156), 
each  side  of  the  equation  =  p. 

163.  Schol.  2.—  The  equation  (Arts.  83  and  91) 

p  cos  0  —  /?,  cos  #!      />  sin  0  —  p\  sin  0j 
/»!  cos  0X  —  /f>2  cos  02      /G!  sin  0£  —  />2  sin  03 

or,          )0  [/>!  sin  (0  -  0X)  -  p2  sin  (0  —  0,)]  =  —Plp2  sin  (^  —  02) 

is  the  polar  equation  of  a  line  through  two  given  points  (plt  0^  and 
0>2,  ^,). 

Proposition  IS. 
164:.  Theorem.—  The  equation 

A     cos  #  +  .£>  sin  6  +  C=0 


represents  a  right  line;  in  which    p    and  0  are  the  polar 
co-ordinates  of  any  point  in  the  line. 

For,  transforming  to  rectangulars,  since,  by  Art,  83,  x  =/>  cos  0, 
and  y=/?  sin  6,  the  equation  becomes  Ax  +  By  +  C  =  0,  which 
is  the  equation  of  a  right  line  by  Art.  109. 

165.  Schol.  1.  —  Ap  cos  0  +  JBp  sin  O+  C—0  can  be  reduced  to 
the  form  p  cos  (0  —  a)  =p  ;  for  by  Art.  142, 

-A  -B  C 

-c  S 


is  of  the  form   p  (cos  a  cos  0  +  sin  a  sin  0)  =/>. 

^ 

.  *  .     p  cos  (0  —  a)  =p,  when  tan  a  =  —  ,  and  |?  = 


A  '  /'  + 


62  RIGHT  LINE. 


166.  Schol.  2.  —  The  condition  that  three  right  lines  given  by  their 
polar  equations,  pass  through  one  point  is  found  in  Art.  152,  and  the 
area  enclosed  by  three  right  lines  in  Arts.  54  and  153. 

167.  Exercises.—  Let  the  vertices  of  a  triangle  be  (xlt  yj,  (x2,  y2), 
and  (#3,  ?/3). 

(1.)  Find  the  equations  of  its  sides  (Art.  90). 

(2.)  Show   (Arts.  37  and   128)  that  the   equations  of  the  perpen- 
diculars which  bisect  its  sides  are 

\ 

(x,  -x2)x  +  fa  -y^y  =  \  (x?  -  *22)  +  1  0/i2  -  y!) 
(x2  -  xj  x  +  fa-y3)y  =  ji  Ctf  -  *32)  +  i  W  ~  2/32) 

(x3  -xjx  +  fa  -y^y=\  (x?  -  xf)  +  \  fa2  -  y,2). 


(3.)  Show  (Art.  116)  that  these  three  perpendiculars  pass  through 
one  point,  and  find  the  co-ordinates  of  the  point  of  intersection. 

(4.)  Show  that  the  equations  of  the  lines  through  the  vertices  and 
perpendicular  to  the  sides  opposite  them  are 

(xl  -x.2)x  +  fa  -  y2)  y  =  (xl  -  x2)  xz  +  fa  -  y2)  y3 
(x2  —  x3~)  x  +  fa  —  y3)  y  =  (x*  —  a?3)  ^i  +  fa  —  2/s)  y\ 
(x^  —  x^  x  +  fa  —  yj  y  =  (x9  —  xj  x2  +  fa  —  yi)  y*. 


(5.)  Show  that  these  three  perpendiculars  also  pass  through  one 
point,  and  find  its  co-ordinates. 

(6.)  Show  (Art.  90)  that  the   equations  of  the   lines  through  the 
vertices  and  bisecting  the  sides  opposite  them  are 


fa  +  y3  —  %0  x  —  (x2  4-  xs  — 
fa  +  yi  —  %)  a;  —  (a?s  +  ^i  - 

(7.)  Show  that  these  three   bisecting  lines  pass  through  a  single 
point,  and  that  its  co-ordinates  are 


EXERCISES.  63 


k    (8.)  Show  that  the  equations  of  the  three  bisectors  of  the  angles  of 
a  triangle  are 

(x  cos  «i  +  y  sin  av  —  p^  —  (x  cos  a2  -f  y  sin  a2  —  p.2)  =  0 
(#  cos  «2  -f  ?/  sin  a2  —  p.2)  —  (x  cos  a,  +  y  sin  ££,  —  £>3)  =  0 
(a;  cos  «3  +  y  sin  a3  —  p3)  —  (x  cos  «i  +  y  sin  a:  —  pi)  =  0. 


(9.)  Show  that  these  bisectors  intersect  in  a  point,  and  find  the  co- 
ordinates of  the  point. 

(10.)  Show  how  many  of  the  points  mentioned  in  (3),  (5),  (7)  and  (9) 
are  upon  the  same  straight  line  (Art.  91). 

(11.)  Show  (Arts.  146  and  148)  that  when  a  line  cuts  the  sides  of  a 
triangle  ABC  (produced  if  necessary)  in  the  points  Jj,  J/and  JV,  then 

AN    £L     CM          j 

NB  '  LC  '  MA  ~~ 

y.  B.  —  Such  a  line  as  LN  is  a  transversal. 

(12.)  Show  (Art.  31)  that  when  lines  be  drawn  from  any  point 
through  the  vertices  of  the  triangle  ABC,  and  meeting  the  opposite 
sides,  EC,  CA,  and  AB  respectively  in  D,  ^7  and  F,  then 

AF    BD    CE  = 
FB'  DO'  EA 

N.  B.  —  The  three  lines  meeting  at  a  point  are  convergenis. 

(13.)  Show  (Art.  151)  that  the  polar  co-ordinates  of  the  intersection 
of  two  right  lines  are 


_  P-*  ~  2P&* cos  ai~ 

Pt  ~  sin  («!  —  a2) 

pl  COS  «2  —  pz  COS  «! 

.'  sin  az—2  sin  a/ 


CHAPTER   IV. 


THE  CIRCLE. 

168.  Tangent  Line.  —  If  a  secant  line  be  passed  through  two 
points,  P2  and  P3,  of  a  curve,  and  the  points  be  conceived  to 
move  continuously  along  the  curve  until  they  meet  at  Pl  (the 
secant  continuing  to  pass  through  the  points  in  all  their  positions), 
the  secant  in  its  limiting  position  is  called  a  tangent  to  the  curve 
at  Plf  and  is  said  to  touch  it  in  two  consecutive  or  coincident 
points  at  Pr 

The  equation  of  the  secant  or  chord 

P2P3  is   (Art.  90),    £=J!l  =  ZL=yim 

x  -  x*      x3-x.2 

"When  the  value  of  the  second  member 
of  this  equation  is  determined  from  the 
equation  of  any  particular  curve,  and 
substituted,  and  the  points  are  then 
made  consecutive,  the  equation  becomes 
that  of  the  tangent  to  the  curve*  The 


intercept  on  x  is  OAl  =  a^.     The  subtangent  is 
a*  —  XL 

The  length  of  the  tangent  is 


,  and  its  length  is 


=  ~\/y\  +  (&i  —  #i)2- 


169.  Normal  Line.  —  A  normal  line  to  a  curve  is  so  situated  that 
it  is  perpendicular  to  the  tangent  at  the  point  of  contact. 

The  equation  of  the  normal  can  be  obtained  from  that  of  the  tangent 
by  finding  the  equation  of  a  line  through  the  point  of  contact,  and 
perpendicular  to  the  tangent  (Art.  128).**  The  intercept  on  x  is 

*  In  Differential  Calculus  the  general  equation  of  the  tangent  line  for  all  curves 


-  —  =  —  —  ' 


*  *  The  general  equation  of  the  normal  line  is  —    —  =  —  —  —  . 

X  —  3/1  ^/l 


64 


GENERAL  EQUATION  OF  THE  CIRCLE. 


65 


OA2  =  <%]  the  subnormal  is  A2J),  and  its  length  is  ^  —  a2.     The  length 
of  the  normal  is  PiA2  =  ]/yia  4-  (xl  —  a.;)8. 


Proposition  1. 
170.  T7ieorem.—The  equation 


represents  a  circle;  in  which  x  and  y  are  the  rectangular 
co-ordinates  of  any  point  of  the  circumference,  Xl  and  yl 
those  of  the  centre,  and  r  is  the  length  of  the  radius. 

For,  by  Art.  26  in  the  equation 


r  represents  the  distance  between 

(xlt  y,}  and  (x2,  y2).     Let  (x2,  y2) 

be  every  point  at  the  distance  r 

from  the  fixed  point  (xlt  y^  ;  then 

if  x   and   y    represent    the    gen-    _ 

eral    values    of  x2    and    y2,    the 

equation  becomes  (x  —  xl)2+  (y  —yl)2  =  r2.     But  the  point  (x,  y)  is 

everywhere  at  the  distance  r  from  (#„  yj,  and  hence  must  be 

always  in  the  circumference  of  a  circle  whose  radius  is  r.     The 

same  equation  can  be  proved  directly  from  the  figure. 

171.  Schol.  1.  —  If  the  origin  be  at  the  centre  then  xl  =  0,  and  yl  =  0, 


If  the  origin  be  on  the  circumference,  we  have  x*  -f  ?/,2  =  r2  ; 
substitute  .  •  .     x*  -f  y2  —  2x&  —  %,y  =  0. 

If  in  addition,  the  axis  of  x  pass  through  the  centre,  then  y^  =  0, 
and          .  •  .     #2  +  y2  —  2x^x  =  0  ;  or  since  now  xl  =  r,  by  transposition, 


If  the  axis  of  y  pass  through  the  centre,   x^  =  0. 
x2  -f  y2  —  2y^y  =  0;    or  since  now  yl=r,      . 


x2  =  2ry  —  y2. 


These  are  all  of  the  form    x2  +  y2  +  Ax  +  By  +  (7=  0. 

5 


THE  CIRCLE. 


172.  Schol.  £.—  Conversely  ,  the  equation  x2  +  y2  +  Ax  +  By  +  C=  0, 
referred  to  rectangular  axes,  always  represents  a  circle  ;  for  it  may  be 

A 

written 


4 

which  is  of  the  form         (x  —  xtf  +  (y  —  y^f  =  r2, 

A  B 

in  which          xl  =  --  ,         and  yl  =  —  —  -' 


If      A*  +  £2-4C>0,     the  circle  has    r  =  ~  \/  A*  +  BL-4C, 
and  the  point  (  —  —  ,  —  —  )  for  its  centre. 

If  A1  +  £2—4C=  0,  the  circle  is  the  point  (xlt  yj. 
If  A2  +  _S2  —  JfC<i  0,  the  circle  is  imaginary. 


173.  Examples.  —  Construct  the  circles  represented  by  the  follow- 
ing equations. 

(1.)    •         x*  +  f-4x  =  5. 

(2.)  x*  +  y2  +  6x  —  6y  =  —  9. 

(3.)  x  -y  —  x2-y2  =0. 

(4.)  x*  +  y*  +  Xy+3y  =  t 

Find  the  radii  and  co-ordinates  of  the  centres  of  the  above  circles. 

174.  Exercise.  —  Show  that  the  equation 

x2  +  yl  +  2xy  cos  yx  +  Ax 
represents  a  circle  referred  to  oblique  axes. 

Proposition  £. 

175.  Theorem.—  The  equation 


expresses  the  rectan/le  of  the  segments  into  which  any  line 
passing  through  a  given  point,  and  cutting  a  given  circle, 


LINE  AND   CIRCLE. 


67 


is  divided  by  that  point ;  in  which  (x,  y)  is  the  given  point, 
p2  the  area  of  the  rectangle,  and  (x  —  x^  +  (y  —  y^f  =  r2 
represents  the  given  circle. 

1st.  Let  P  be  without  the 
given  circle. 

If  PPl  =  d}  then  (x—xtf  + 

equation  of  the  circle  through 
P  with  centre  Pr    If  d2  =  r2  + 
p2,  then£>  is  the  length  of  the     . 
tangent  from  P  to  the  circle 


.  '  .     Substituting,  eq.  (a.)  becomes  (x  —  x^2  +  (y  —  3/i)2  =  r2  +  p2. 
But,  by  elementary  geometry,  p2  =  PSl  .  PS2. 

/S^  and  S2  are  in  the  same  direction  from  P,  and  the  rectangle 
.  PS2  is  therefore  positive. 


2d.  Let  P  be  within  the 
given  circle. 

If  PPl=d,t'he.u(x-xl)2  + 
(y-yl)2  =  d2  .  .  .  (6.)  is  the 
equation  of  the  circle  through 
P  with  the  centre  Pr  If 
d2  =  r2  —  p2,  then  p  is  the 
length  of  the  tangent  from 
P  to  its  intersection  with  the 
circle  (x  —  x^2  +  (y  —  y^f  =  r2. 

.'.     Substituting,    eq.  (6.)  becomes    (x  —  xJ2+  (y  —  yl)2  =  ^—  p2. 
But,  —p2  =  PSl  .  PS2,  by  elementary  geometry. 

Sl  and  $j  are  in  opposite  directions  from  P;  the  rectangle  of 
P&!  and  P£j  is  therefore  negative. 

176.  Example.  —  Find  the  rectangle  of  the  segments  of  the  secant, 
drawn  from  the  point  (  —  2,  5),  cutting  the  circle  4  —  ^2  —  -^3/2  =  x- 

Am.  26. 


68 


THE  CIRCLE. 


Proposition 
177.  Theorem.—  The  equation 


represents  a  right  line,  called  the  axis  radical  of  the  two 
circles  whose  radii  are    r-^    and    r2,   and  whose  centres  are 

at    (xi,  2/1)    and    (x2,  y^)    respectively. 

For,  expand  and  cancel, 
and  we  have, 


which  (Art.  109)  is  the 
equation  of  a  right  line. 
If  some  point  P  —  i.  e., 
(x,  y)  —  be  taken  on  this 
line  without  both  circles, 
then,  by  Art.  175, 


Y  +  (y  -  y,)2  -  n2  =  (x  -  *2)2  +  (y  -  ytf  -  r*  =/, 

which  expresses  the  property  of  the  axis  radical,  that  the  tangent 
drawn  from  any  point  of  the  line  to  one  circle,  is  equal  to  the 
tangent  drawn  from  the  same  point  to  the  other  circle.  The  line 
is  real  whether  the  circles  intersect  or  not;  and  if  it  passes 
within  one  circle,  it  passes  within  the  other  at  the  same  time. 

178.  Schol.  1.  —  The  equation  to  the  line  of  centres  is  (Art.  90) 


The  equation  to  the  axis  radical  is 


—  xl 


.  x+  C, 


hence  these  two  lines  are  perpendicular  to  each  other. 

179.  Schol.  2. — The  equations  of  the  axes  radical  of  three  circles 


TANGENT  LINE. 


69 


taken  two  and  two  whose  centres  are  the  points  fa,  y^),  (x2,  y2)>  (£3,  3/3), 
are  (Art.  177) 

(x,  -xjx  +  (y,  -  y,)  y  =  }  [fo*  -  xfi  +  (yx2  -  y22)  -  (n2  -  r22)] 
(*,  -  a,)  a?  +  (y,  -  y,)  y  -  f  [(*22  -  *.')  +  (y22  -  y32)  -  (r22  -  r32)] 
(0:3  -  ar,)  *  +  (y,  -  y,)  y  =  j  [(*32  -  xf)  +  (y32  -  ^)  -  (^  ~  rfi] 


which  intersect  in  a  single  point  called  the  (Art.  116)  centre  radical. 
Compare  these  equations  with  the  equations  of  the  parallels  to  them. 
Art.  167  (2).  This  may  also  Tbe  more  easily  demonstrated  as  follows  : 
if  by  the  equation  S\  =  0  be  understood  (x  —  a^)2  -f-  (y  —  y^2  —  r*  =  0, 
then  will  Si  =  0,  /S^  =  0,  S*  =  0,  represent  three  circles, 
and  the  equations  Si  —  S2  =  0,  ^  —  ^  =  0,  S3  —  /Si  =  0, 

will  be  those  of  their  axes  radical,  which  meet  in  a  single  point 
(Art.  116). 

180.'  Example.  —  Find  the  axes  radical  of  three  circles,  two  by 
two,  whose  radii  are  respectively  2,  3  and  ^,  and  whose  centres  are  at 
the  points  (1,  #),  (5,  —  2}  and  (—  1,  —  3)  ;  and  show  that  the  three 
intersect  in  a  common  point.  Ans.  8x—  8y  —  19. 

y=    7. 

y  =  13. 

Proposition  4. 
181.  TJieorem.  —  The  equation 


represents  a  line  tangent  to  the  circle    x*  +  2/2  =  r2 ;  in  which 
(2*1 2/0    is  the  point  of  tangency. 

For,  if  two  points  P2  and  P3  be 
taken  upon  the  circumference  of 
the  circle  x2  +  y2  =  r2,  the  equations 
•Co  i  t/9  —  I/  ctnci  *t/o  "™i  ?/o  —  /  cir\3 
the  equations  of  condition  that  P2 
and  P3  be  so  situated. 

The  equation  of  the  line  P2P3  is  (Art.  90)  y—~^-  =^ 


70  THE  CIRCLE. 


But 


z3-*2      2/3  +  2/2 

Substituting  this  value  in  eq.  (a.}  we  have 


as  the  equation  of  the  line  secant  to  the  circle  xz  +  y2  =  r2,  through 
P2  and  P3.  Let  P2  and  P3  be  conceived  to  approach  each  other 
along  the  curve  until  they  are  consecutive  points  at  Pl  (Art.  168). 
The  secant  line  passing  through  these  consecutive  points  will  be 
the  tangent,  and  we  shall  have,  x3  =  x2  =  x1}  and  y3  =  y.2  =  yr 

ij  —  11  x 

.'.   eq.  (6.)  becomes      —  r  =  —  -7,    or    Wi—'yJf^-XBi  +  'xf. 

ju      a^  yl 


182.  Cor.  1.  —  If   the  origin   be   changed  (Art.  23),  the  equation 
becomes  (x  —  XQ)  fa  —  x0)  +  (y  —  y0)  (yr  —  y0)  =  r2,  which  is  tangent  to 
the  circle  (x  —  #0)2  +  (y  —  yo)2  =  ^2,  at  the  point  fa,  y^}. 

183.  Cor.  2.  —  The  intercepts  of  the  tangent  upon  the  axes  are 


184.  Cor.  3.—  The  subtangent 

T<t  +  yo  (y\  —  yo)  —  fa  —  ^o)2 


T2  y.2  _  x  2 

When  the  origin  is  at  the  centre,  the  subtangent  =•     —xl  =  —      —. 

Xi  Xl 


PARALLEL   TANGENTS   TO  A   CIRCLE.  71 

185.  Schol. — The  equation  of  the  tangent  at  (xv,  ?A)  (see  fig.  of  Art. 
187)  is 

x\x  +  y\y  —  r*, 

and  that  of  the  tangent  at  the  other  extremity  of  the  diameter — i.  e.,  at 
(— tfi,  —2/i)— is 


.  • .     xlx  +  yly  =  ±r*,ory  +  —  x  =  ±r- 

2/i  2/i 

represents  the   tangent  through  (x^  yj,  and   also  the  only  tangent 
parallel  to  it. 

Let    XOPl  =  Olt  then  —  =  tan  ^  =  m,  v '/  , 

2/1  /&*,  £ 

<\  f 


and  by  trig.     —  =  sec  9l  =  -/tan  ^+1  =  j/w'  +  A      f  • 


tf, 


' 


.'.     y  +  ra;r  —  ±  ryy&  +  ./, 
or  when  the  origin  is  not  at  the  centre  (Art.  23), 

y  —  y0  +  m  (x  —  x0)  =  ±  r  ^/m*  +  ^ 

represents  the  parallel  tangents  to  the  circle  (a;  —  ar0)a  +  (y  —  2/o)2  =  ^ 
and  is  often  called  their  magic  equation. 

186.  Examples.  —  Form   the    equations   to  the   following  tangent 
lines. 

(1.)  Circle  x1  +  y*  =  9,  at  the  point  (2.4,  1.8). 

Ans.  3    +  4x  =  15. 


(2.)  Circle  x2  +  /  —  ./Or  —  12y  =  —  57,  at  the  point  (4£,  4-06). 


(3.)  Form  the  equations  to  the  tangent  lines  parallel  to  those  in 
examples  (1)  and  (2). 

•  (4.)  Find  the  intercepts  of  the  tangents  in  examples  (1),  (2)  and  (3), 
and  their  subtangents. 


72 


THE  CIRCLE. 


Proposition  5. 

187.  Theorem.— The  equation 

represents  a  right  line  normal  to  the  circle   x*  +  y2  =  r2,    at 
the  point    fa,  y^). 

Y 

For,  resuming  the  equation  of 
the   line   tangent  to  the   circle 
=  r2     at  P1;  viz. : 

'U 'Ui  'JG* 

(Art.  181) 


y—y\       xi 


—  x. 


by  Art.  123,  ^—^  =  +  ^ 

x—xv          xl 

is  perpendicular  to  the  tangent  at  Pl ;  it  is  therefore  the  normal. 

Since  this  line  passes  through  the  origin,  all  normals  to  the  circle  pass  through 
its  centre. 

188.  Cor.  1. — If  the  origin  be  changed,  the  equation  becomes 


which  is  evidently  the  equation  of  a  line  through  Pl  and  the  centre 
P0,  and  is  normal  to  the  circle  (x  —  x0)2  +  (y  —  3/0)2  =  r2. 

189.  Cor.  2. — The  intercepts  of  the  normal  are, 
when    y  =  0,          x  =  — *^ —  =  a 


when    x  =  0,         y  = 


The  subnormal     =a-x,= 


190.  Example.  —  Form  the  equation  of  the  line  normal  to  the  circle 
4x  +  y  +  2  =  OtoA,  the  point  (4,  5).        Ans.  4y  +  24  =  llx. 


CENTRES  OF  SIMILITUDE. 


73 


Proposition  tf. 
191.  Theorem.— The  equations 


j 

and      = 


express  the  positions  of  the  intersections  of  the  common 
tangents  of  two  circles— i.  e.,  their  centres  of  similitude;  in 
which  ?*!  and  r2  are  the  radii,  and  (xl}  yi),  (x2,  #2)  are 
the  centres  of  the  circles. 

By  geometry  S  and  Sl  fall  on  C2Ci. 


M 


A, 


By  similarity  of  triangles 
r :  r : : 


.    c^    .  ov* 

.  .    OOj     .  OL/2 

:  :  MA,  :  MA2.      . 
.  * .     solving  x8 


r1-r2 


and  similarly, 


74  THE  CIRCLE. 


Again, 


rx 
and  solving,  x8 


.,   , 

and  similarly, 


s 

Fj  T  ^2 

Compare  the  equations  of  Art.  37. 

192.  Cor.—  Draw  any  parallel  radii,  as  C(P,  C2Pr,  then  PP'  cuts 
in  some  point  Q  —  i.  «.,  (a:2,  ya). 

.   .  '  .     rl:ra::QP:QPt 


n  —  r2 
.  •  .    "2?,=  a:a,  and  similarly  ya  =  yr 

The  same  may  be  proved  respecting  Si.  Hence  all  lines  drawn 
through  the  extremities  of  parallel  radii  pass  through  S  or  /Si,  and  are 
cut  by  the  circles,  so  that  rt  :  r2  :  :  SP  :  8Pr.  For  this  reason  S  is 
called  the  external  and  /Si  the  internal  centre  of  similitude. 

193.  Schol.  1.  —  It  is  possible  to  obtain  the  equations  of  the  four 
common  tangents  to  the  two  circles  (x  —  #i)2  +  (y  —  y^f  =  r*,  and 
(x  —  a?2)2  +  (y  —  y^  —  r<?  from  Art.  185,  by  finding  the  four  values 
of  m  to  be  obtained  from  the  conditions  m  =  ml^m2  and 


*  -f 

and  then  in  the  general  equation  of  the  tangent  to  each  circle,  each  of 
which  is  of  the  form,  y  —  yQ  +  m  (x  —  XQ)  =  ±  r  •\/'rr?  +  -^,  substituting 
these  values  :  equations  will  thus  be  obtained,  four  of  which  will  be 
identical. 


FOUR   TANGENTS  COMMON  TO   TWO  CIRCLES.  75 

E.  G.t  The  four  common  tangents  to  the  circles 

(y-£)2  +  0-£)2=0 
(y-iY  +  *?  =  4 
are  obtained  by  the  general  equations  (Art.  185), 

y  —  2  +m(x  —  5)  =  ±3  -j/W  +  1  .  .  .  (a.) 


becoming  identical. 

Eliminate  x  and  y  from  («.)  and  (6.),  and  we  find  the  four  values 

m  =  24,    m  =  oo,    m  •=  --  ,    m  =  0t 

-Z<V 

and  (a.)  becomes,  by  substituting  these  four  values  of  m, 
y  -f  2.4x  =  6.2,  or  =  21.8 


5         19  10 

y-^^'or=-y 

y  =  —  1,  or  =  5 
and  (6.)  becomes  in  a  similar  manner, 


19 


y  =  —  1,  or  =3. 

Hence  the  four  common  tangents  are, 
y  +  2.4x  = 


76 


THE  CIRCLE. 


191.  Schol.  2. — The  external  centres  of  similitude  of  three  circles, 
taken  two  and  two,  lie  on  one* line  called  the  axis  of  similitude.  For 
the  centres  of  similitude 


tr^—r^c^  rt?/2  —  r2yA 

\   rl  —  r2  n  —  r2    J 

(r&s  —  r&^  r*y*—r*y\ 

r2  —  r3  r2  —  r3    J 


r3  —  rx 


will  be  found  by  substitution  to  fulfill  the  equation  of  condition  of 
Art.  91. 

There  are  also,  as  may  be  proved  in  like  manner,  three  internal  axes 
of  similitude,  on  each  of  which  falls  one  external  and  two  internal 
centres  of  similitude.  Construct  the  figure  by  finding  the  intersections 
of  the  common  tangents  of  three  non-intersecting  circles  and  drawing 
the  axes  of  similitude. 


Proposition  7. 
195.  Theorem.—  The  equation 


represents  a  circle  with  radius  r;    in  which  p   is  the  radius 
vector  of  any  point  of  the  circle,  and  pt  that  of  the  centre. 

By  trigonometry, 

Or  if  the  axis  of  #  is  the  initial 
line,  we  may  write  since 

P\~  P\     x~~  x~  x  —  "  ~a> 


or,  p2  -  %>!  cos  (6  -  a)  =  r*  -  p?.  .  .  (b.) 

This  equation  may  also  be  obtained  by  transformation. 


EXERCISES. 


196.  Schol. — The  general  form  of  the  polar  equation  of  the  circle  is 

p*  +  Ap  cos  0  +  Bp  sin  0  -f  C=  0, 
for,  by  trig,  we  may  write  eq.  (6.) 

p2  —  2ppi  (cos  0  cos  a  +  sin  0  sin  a)  —  (?*2  —  /o/2)  =  0, 
or,      /o2  —  (2pi  cos  a)  />  cos  0  —  (£/>!  sin  a)  p  sin  0  —  (r2  —  p2)  =  0. 
Now  place  the  constants, 

— .2pi  cos  a~  A,       —  2pi  sin  a  =  J5,  and       —  (r2  —  jO^)  =  (7; 
.  * .  /o2  +  ^4/>  cos  0  +  Bp  sin  0  +  C=  0. 

197.  Cor. — If  the  pole  be  upon  the  circumference,  PI  =  r, 

p2  =  2pr  cos  ^.,  which  has  the  two  roots,   p  —  0,  and  p  =  2r  cos  ^. 

The  latter  is  then  the  polar  equation  of  the  circle. 

If,  however,  the  pole  is  at  the  centre,  pi  =  0,     and    />2  =  r2, 

.  •  .     p  =  r  is  then  the  polar  equation  of  the  circle. 

198.  JSteeretees.— (1.)  Prove  that  the 
perpendicular  "from  the   origin  upon  the          j) 


tangent       p= 


(2.)  Show  (Art.  172)  that  the  equation 


X 


represents  some  circle  through  (&lt  yi). 

Also  find  the  equation  when  A  or  B  is  eliminated  instead  of  0. 

.y  Show  that  the  equation 

(x*  +  f)  (xiy.t  -  x,yi}  +  C(yi  -  y,)  x 

+  (X2  + 


represents  some  circle  through  the  two  points  (xit  ?/0  and  (a^,  y2). 
Also  when  5  and  C  are  eliminated  the  same  equation  is 

(x*  +y2)  (yi  -yt)  +  (^  +2A2)  (y,  -y)  +  (^2  +  yf)  (y-yO 
-  ^[[(ar  —  x^  2/2  +  (a;2  —  x)  yi  +  (^i  —  x^  y]  =  0. 


78 


THE  CIRCLE. 


(4.)  Show  that  the  equation 

(x*  +  y2)  [fe  -  *,)  y3  +  (a;,  -  *,)  y2  +  (arf  -  a*)  y,]  ^ 
—  (#i2  +  yi2)  [(a?2  —  a?8)  y  +  (a;  —  #2)  y3  +  (a;3  —  a;)  yj 
+  (^22  +  y22)  [(a*  —  ^)  yi  +  (a?i  —  a;8)  y  +  (a;  —  ^)  ys] 
-  fe2  +  y32)  [(a?  —  arj)  y2  +  (a?»  —  x)  yl  +  fe  —  #2)y]  ^ 

represents  a  circle  through  the  three  points  (a^,  yO,  (x2,  y2)  and  (z3,  y3). 
Also  show  that  the  equation  of  condition  for  four  points  to  be  upon 
the  same  circumference  is  expressed  by  substituting  (x4,  y4)  for  (x,  y) 
in  this  equation. 

(5.)  Show  (Art.  31)  that  the  equation  of  condition  of  four  points  on 
one  circumference  (4)  has  this  geometric  significance  : 


when  /»!,  p2,  p3,  /?4  are  the  distances  of  the  points  Plt  P2,  P3,  P4  on  the 
same  circumference,  and  Pl  P2  P$  signifies  the  area  of  the  triangle, 
Pi  P2  P3,  etc. 

(6.)  Show  that 

x*  +  y2  —  («!  +  03)  a;  +  G^OJ  +  By  =  0 
represents  the  circle  whose  intercepts  on  the  axis  of  x  are  Oi  and  az. 

(7.)  Show  that 


represents  the  circle  whose  intercepts  on  the  axis  of  y  are  bi  and  b2 
(8.)  Show  that 


represents  the  circle  whose  intercepts  on  the  axes  are  a1(  a2,  ^  and  b2. 

(9.)  Find  the  conditions  that  a  circle  touch  each  and  both  of  the 
axes. 


CHAPTER   V. 

EQUATIONS   AND   LOCI. 

199.  A  plane  locus  is  the  line,  straight  or  curved,  described  by 
a  point  moving  on  a  plane  according  to  some  law. 

This  law  consists  in  the  existence  of  &  fixed  relation  between 
the  x  and  y  co-ordinates  of  the  point,  in  all  its  successive  posi- 
tions. An  equation  of  a  locus  is  used  to  express  this  law. 

200.  The  connection  between  a  locus  and  its  equation  is  the 

fundamental  idea  of  co-ordinate  geometry. 

E.  G.  It  was  shown  (Art.  109)  that  the  locus  of  a  point  moving 
according  to  a  law  expressed  by  any  equation  of  the  first  degree  in 
x  and  y  is  some  right  line.  Thus,  in  the  equation 

Ax+By+C=0,     let  A=l,  B=l     and  C=2,     .'.     x  +  y  +  2  =  0, 

if  any  value  whatever  be  assigned. to  x,  a  corresponding  value  of  y  will 
be  obtained  from  the  equation ;  and  the  points  obtained  by  assigning 
successive  values  to  x  will  be  found  to  be  all  situated  upon  a  fixed  right 
line. 

201.  Variables. — The  co-ordinates  x  and  y  are  called  variables, 
or  sometimes  current  or  running  co-ordinates. 

202.  Constants. — The  quantities  A,  B  and  C  are  called  con- 
stants.    When  given,  they  cause  the  locus  to  be  definite  and  fixed. 
If  varied,  they  move  the  locus  to  some  new  position  or  effect  some 
change   of  form.      When  unknown  or  undetermined,  they  are 
sometimes   called    variable   parameters  or    arbitrary   constants. 
Constants  are  co-efficients  or  exponents  of  variables. 

79 


80  EQUATIONS  AND  LOCI. 

203.  The  equation  of  a  locus  is  composed,  then,  of  two  kinds  of 
quantities,  variables  and  constants. 

Variables  are  the  co-ordinates  of  a  moving  point,  as  (x,  y), 
(ft,  6),  etc.,  for  general  values ;  (xl}  yj,  (plt  OJ,  etc.,  for  restricted 
values  (Art.  19). 

Constants  are  the  co-ordinates  of  a  locus. 

JV.  jB. — We  have  found  it  convenient  to  use  A,  B,  C,  m,  p,  a,  b,  c, 
etc.,  for  general  values  of  constants,  and  Alt  Bl}  Ci,  m^  p^  etc.,  for 
restricted  values.  Also,  xlt  y1}  plt  Ot,  etc.,  are  frequently  considered  to 
be  constants. 

204.  Constants  impose  the  restrictions  to  which  variables  are 
subject,  and  the  conditions  which  a  locus  may  fulfill. 

205.  A  Condition  is  some  fixed  relation  existing  between  two 
or  more  loci.     An  equation  of  condition  is  used  to  express  this 
relation. 

E.  G.,  It  was  shown  in  Art.  123  that  mi  = is  the  equation  of 

w2 

condition ,  which  is  true  whenever  the  lines  y  =  m^x  +  b^  and  y  =  m.2x  +  b.z 
are  perpendicular  to  each  other. 

206.  Transformation  of  Co-ordinates   is  the  reference  of  any 
fixed  or  moving  point  to  a  new  system  of  co-ordinates. 

207.  Equations  of  Transformation  express  the  relation  between 
the  primitive  co-ordinates  and  the  new,  and  contain  new  vari- 
ables, primitive  variables  and  constants. 

208.  Equations  of  numerical  value  are  for  the  purposes  of  com- 
putation.    When  an  equation  of  any  kind  becomes  determinate, 
it  is  of  this  character,  as,  moreover,  are  the  equations  of  ordinary 
algebra. 

209.  Cyclic   symmetry  exists   in   an   expression   when   certain 
letters  or  suffixes  are  exchanged  or  permuted  in  a  cycle  (see  Art. 
34),  to  form  its  different  terms  or  equations. 


DIMENSIONS.  LOCUS.  81 

E.  G.,  In  the  equation  of  Art.  34  the  suffixes  are  permuted  in  a 
cycle  of  three.  In  the  equations  of  Arts.  167  and  198  a  kind  of  double 
cyclic  symmetry  is  noticeable.  A  symmetric  arrangement  is  a  useful 
aid  to  the  memory,  and  greatly  facilitates  algebraic  work. 


Proposition  1. 

210.  Theorem.— Every  term  of  any  equation  whatever  rep- 
resents  quantity  of  the  same  kind,    be   it  volume,    area, 
distance,  mere  number  or  other  measurable  thing. 

For  the  terms  are  connected  by  one  of  the  signs  +  ,  — ,  =;  and 
since  it  is  impossible  to  increase  or  diminish  anything  except  by  a 
quantity  of  the  same  nature,  or  to  affirm  the  equality  of  unlike 
things,  the  proposition  must  be  true. 

211.  ScJiol. — A  volume  is  said  to  be  of  three  dimensions,  an  area  of 
two,  a  distance  of  one,  and  a  ratio,  or  mere  number,  of  no  dimensions. 

Space  is  at  most  of  only  three  dimensions,  but  space  of  four,  five 
or  n  dimensions  may  be  used  as  a  purely  analytic  conception. 


Proposition  2. 

212.  Theorem. — Any   single   equation   between    x    and    y 
represents  some  plane  locus. 

For  it  expresses  some  fixed  relation  of  x  and  y,  such  that  if 
x  have  a  given  value,  y  becomes  known.  By  assuming  successive 
consecutive  values  of  xt  and  finding  the  corresponding  consecutive 
values  of  y,  we  trace  the  motion  of  a  point.  A  point  so  moving 
describes  a  locus  (Art.  199). 

213.  Schol. — The  locus  of  an  equation  which  has  no  terms  except 
those  which  contain  x  and  y  passes  through  the  origin.     For  let  one  of 
the  successive  values  of  a;  be   x  =  0,  then  y  =  0 ;    but  these  are  the 
co-ordinates   of  the    origin.      E.  G.,  In  the   equation  if  =  4px   (see 
figure  in  Art.  243),  if  x  =  0,  then  y~-=0.     See  also  Arts.  109  and  171. 
It  will  hereafter  be  shown  (Art.  458)  that  when  in  addition  there  are 
no  terms  of  the  first  degree  the  locus  passes  twice  through  the  origin ; 


82  EQUATIONS  AND  LOCL 

and  when  there  are  no  terms  of  the  second  or  lower  degrees,  the  locus 
passes  three  times  through  the  origin,  etc. 

214:.  Examples. — Show  the  truth  of  the  above  proposition  numeri- 
cally by  constructing  the  loci  represented  by  the  following  equations. 


x  —  3  x—1 

(2.)         y  =  — 


Proposition  3. 

215.  Theorem.— Two  simultaneous  equations  between  x  and 
y,    one  of  the    mth    degree  and  the  other  of  the    nth    degree, 
represent,  in  general,     mn     definite  points  upon  a  plane, 
which  are  the    mn    intersections  of  the  loci  of  the  two  equa- 
tions, considered  singly,  as  in  Art.  212. 

For,  the  two  equations  represent,  each  singly,  a  locus.  The 
values  of  x  and  y  can  be  simultaneous — that  is,  can  refer  to  the 
same  points,  only  at  the  points  of  intersection  of  the  two  loci. 
If  the  x  and  y  of  both  equations  are  the  same,  the  two  equations 
will  be  sufficient  to  find  by  elimination  the  definite  values  of  x 
and  y,  which  are  the  co-ordinates  of  the  points  of  intersection. 
But  by  algebra  the  degree  of  the  equation  resulting  from  elimi- 
nation between  two  equations,  one  of  the  mth  and  the  other  of 
the  nth  degree,  is  mnt  and  such  an  equation  will  have  mn  roots. 
There  will  therefore  be  mn  definite  values  of  x  and  y,  and  mn 
intersections  of  the  two  loci. 

216.  Schol.  1. — To  find  the  intercepts  of  any  locus  on  the  axis  of  x, 
let  y  =  0,  which  is  the  equation  of  the  axis  of  x.     If  x  =  0,  we  shall 
obtain  the  intercepts  on  y. 

217.  Schol.  2. — Any   number   of  these  mn  points  may   coincide, 
which  will  be  indicated   by  a  corresponding   number  of  equal  roots. 
Any  number  of  the  mn  points  may  be  at  the  origin,  or  at  infinity,  when 
the  corresponding  roots  will  be  0  or  oo.     Any  even  number  of  the  mn 


INTERSECTIONS  OF  LOCI.  83 

points  may  be  imaginary — that  is,  impossible — which  will  be  indicated 
by  corresponding  pairs  of  imaginary  roots. 

218.  Examples. — Find  the  points  of  intersection  of  the  loci  repre- 
sented by  the  following  equations. 

(1.)  +      =  1 


Ans. 


(2.)  y*  =  y?     and  yi  =  4x. 

(0,  01  (0,  0), 

Ans.   2 


Proposition  4. 

219.  Tfieorem.  —  The  sum  or  difference  of  two  equations 
of  loci,   of  the    m*    and    nth    degrees  respectively,   is,   in 
general,  an  equation  representing  a  locus  passing  through 
the   mn   points  of  intersection  of  the  two  loci. 

For,  if  Sm  =  0  be  understood  to  be  an  equation  of  the  mth 
degree  in  x  and  y,  and  Sn  =  0  one  of  the  nth  degree,  and  k  any 
number,  then  Sm±kSn=0  is  the  equation  of  some  locus,  since 
it  contains  x  and  y;  and  since,  when  Sn  —  0  and  Sn  =  0,  we 
have  Sm  ±  kSn  =  0  also,  the  three  equations  are  simultaneous 
at  the  points  of  intersection  of  Sm  =  0  and  Sn  =  0. 

220.  Examples.  —  Construct  the  loci  represented  by  the  following 
equations,  and  show  that  they  pass  through  the  points  of  intersection 
of  the  loci  represented  by  their  component  equations. 


84  EQUATIONS  AND  LOCI. 


(1-) 

fr+y-Q  +  P  +  tf-Q^O. 

(2.) 

(x  +  y-l)-(x>  +  f-4)=0. 

(3.) 

3(x  +  y-l)-(i?  +  yt-4)=0. 

(40 

(x*  +  y'  +  2x  —  S)  —  2(xt  +  y*  —  8x  +  r)=0. 

Arts.  115  and  177  also  furnish  examples  under  this  proposition. 


^Proposition  &.f 

221.  Theorem.  —  Any  equation  of  a  locus,  some  of  whose 
constants   are   general  (that    is,    their    values    are   unde- 
termined), can,  in  general,  be  made  to  fulfill  as  many  con- 
ditions as  there  are  independent,  undetermined  constants 
in  the  equation. 

For,  a  condition  may  be  expressed  by  an  equation  containing 
some  one  or  more  undetermined  constants,  and  two  conditions  by 
two  or  more  such  equations  ;  one  equation  suffices  to  determine 
one  constant,  or  two  equations  two  constants,  etc.  There  can 
then  be  as  many  conditions  —  that  is,  equations  —  as  there  are  un- 
determined constants. 

222.  Schol.  1.  —  Any  locus  whose  equation  contains  undetermined 
constants  can,  in   general,  be  made  to  pass  through  as  many  given 
points  as  there  are  independent,  undetermined  constants  in  its  equation. 
For,  if  the  co-ordinates  of  one  point,  as  (#lf  y^,  satisfy  the  equation, 
then,  when  xl  and  yl  are  substituted  for  x  and  y  in  the  equation  of  the 
locus,  the  equation  becomes  the  equation  of  condition  that  xl  and  yl 
shall  be  on  the  locus.     Between  this  equation  of  condition  and  the 
equation  of  the  locus,  one  constant  can  be  eliminated.     The  same  pro- 
cess for  a  second  point  would  eliminate  a  second  constant,  and  so  on. 

/£  nt 

223.  E.  G.  Take  the  equation  of  a  right  line      -  +  •-  =  ./,  .  .  .  (a.) 

a      b 

and  subject  it  to  the  condition  of  passing  through  the  points  (xlt  3/1) 
and  (#2,  2/2).     If  (xlt  T/:)  is  on  the  line  we  must  have 


CONSTANTS  AND   CONDITIONS.  85 


and  eliminating  between  equations  (a.)  and  (6.),  we  obtain 

£  +  (a-£l)y  =  z         _ 
a  ay^ 

If  (ar2,  y*)  is  on  the  line,  we  must  have 

^+f  =;,...«*.) 

a        6 
and  eliminating  between  (a.)  and  (cf.),  we  find  similarly 

^(o-^^      ^.^ 
a  ay2 

Now  eliminate  a  between  equations  (c.)  and  (e.),  and  we  have 


as  in  Art.  90. 

224.  ScJtol.  2.  —  Any  locus  whose  equation  contains  undetermined 
constants  can  in  general  be  made  to  be  tangent  to  as  many  given  loci 
as  there  are  independent,  undetermined  constants  in  its  equation.     For, 
the  co-ordinates  of  the  points  of  intersection  of  one  given  locus  with 
this  locus  can  be  found  by  elimination,  in  terms  of  the  constants  of  the 
two  equations.     If  the  condition  be  found  that  will  cause  two  points  of 
intersection  to  coincide  —  that  is,  form  the  condition  that  we  have  equal 
roots  —  the  loci  will  be  tangent  to  each  other.     This  is  evidently  one 
equation  of  condition,  and  will  fix  one  constant.     A  tangency  with  a 
second  locus  will  fix  a  second  constant,  etc. 

225.  E.  G.  Take  the  circle 

(*-3r,)*  +  (y-yi)'  =  r»,  .  .  .  (a.) 

containing  three  independent,  undetermined  constants,  and  cause  it  to 
touch  the  three  right  lines 

y  =  x-2,  .  .  .  (6.) 
=  2,  .  .  .  (c.) 


By  eliminating  bet\veen  equations  (a.}  and  (6.),  we  obtain 


EQUATIONS  AND  LOCI. 


and  the  condition  that  this  quadratic  equation  in  x  shall  have  equal 
roots  is,  by  algebra, 


£ 

whence  we  find 


which  is  the  equation  of  condition  that  the  circle  touch  (6.) 
Proceeding  in  a  similar  manner  with  (c.),  we  obtain 


From  (a.)  and  (d.)  we  obtain 


Equations  (/.)  and  ((/.}  are  the  equations  of  condition  that  the  circle 
shall  touch  (c.)  and  (d.)  respectively.  We  have,  then,  three  equations 
involving  three  undetermined  quantities,  #1(  yl  and  r  ;  hence  by  elimi- 
nation we  can  obtain  such  values  as  will  cause  the  circle  to  touch  all 
three  lines  (5.),  (c.),  (d.)  at  once.  Subtracting  (e.)  from  (/.),  we 
obtain 


The  roots  of  this  equation  are 

xl  =  2    and  y±  =  0. 

This  value  of  Xi  in  (g.)  gives  r  —  2  ;  and  placing  in  equation  (e.) 
xi  =  2    and    r  =  2,    we  find 


Also  if  we  make  y^  =  0  and  xl=4±rm  equation  (e.),  we  find 


Since  a  negative  radius  gives  an  imaginary  circle,  we  can  only  use  the 
upper  sign  of  ± 


,  or  0.828  +  ; 
whence  a?i  =  4  ±  (V    ±  ^)  =  5  .172,  or 


TANGENCIES  OF  LOCI,  ETC.  87 

Thus  it  appears  that  there  are  four  circles  which  touch  the  three  lines, 
as  follows : 

O,l  n,      tf)  ni      O-     /   lV  A«    Q   . 

•**»  ^i        *j  yi  r    /V)  ' 

3d.         ^-^-^v7^,  .Vi  =  0,  r  =  2v'2  —  2; 

4th.        ^  =  6- 

226.  Examples* — (1.)  Find  the    equations  of  condition   that  the 
right  line    y  —  mx  +  b     may  be  tangent  to  the  two  circles 


Am.  b  =  ±  2y'l  +  m\     and    b  =  —  6m± 
(2.)  Find  the  resulting  values  of  b  and  m. 

Ans.  b  =  ±  12  or  T  12  and  m  -  ± 


(3.)  Show  that  the  equations  of  the  four  lines  which  touch  the  two 
circles  are 

yy-^5  =  x  +  12,         and    yy/~ll  =  5x  —  12, 
yy!$5  =  —  x  —  12,     and    y  j/77  =  —  5x  +  12. 

Proposition  6'.t 

227.  TJieorem.—The  general  equation  of  the  nth  degree- 
is.  e.,  an  equation  in  which  all  the  constants  have  general 
values—  can  be  made  to  fulfill  |  n(n  +  3}  conditions. 

For,  the  general  equation  of  the  first  degree  contains  two  con- 
stants independent  of  each  other,  that  of  the  second  degree  2  +  3, 
and  that  of  the  nih  degree  2+3  +  4+  .....  +  (n  +  •  1],  or,  by 
summing  the  series,  ~  n(n  +  3}  constants;  and  hence  (Art.  221) 

can  fulfill  I  n(n  +  3)  conditions. 


EQUATIONS  AND  LOCI. 


228.  Scttol.  —  Some  curve  of  the  second  degree  can  be  made  to  pass 
through  'smy  five  points,  or  be  tangent  to  any  five  given  loci  (Arts.  222 
and  224). 

Proposition  7. 

229.  Theorem.  —  Transformation    of   co-ordinates    cannot 
change  the  form  of  the  locus,  nor  the  degree  of  any  equa- 
tion transformed. 

The/orm  of  the  locus  is  unchanged,  for  that  depends  upon  the 
relation  of  the  different  positions  of  the  moving  point  to  each 
other,  and  not  upon  the  co-ordinates  by  which  relation  is  expressed. 

The  degree  is  unchanged,  for  in  any  transformation  by  equa- 
tions of  the  form 

x  =  ml  x'  +  m2  y'  +  m3 
y  =  nl  x'  +  n2  y'  +  n3 

(Art.  89)  the  values  of  the  new  variables  being  of  the  first  degree 
in  terms  of  the  primitive,  x2,  or  xy,  or  y2  could  at  most  be  com- 
posed only  of  terms  containing  x'2,  x'  y'  ',  x'2,  xf,  yf  ,  etc.  ;  there- 
fore the  degree  of  the  expression  in  the  new  variables  could 
not  be  increased.  Neither  could  it  be  diminished;  for  if  any 
transformation  could  cause  the  resulting  equation  to  be  of  lower 
degree,  then,  by  a  retransformation  to  the  original  expression,  an 
equation  of  higher  degree  would  be  obtained,  which  has  been 
proved  to  be  impossible. 

230.  Example.  —  Kefer    the    locus    represented   by   the    equation 
a:2  +  y*  =  9,  in  rectangulars,  to  new  axes  in  which  yx  =  60°  ,  making  the 
co-ordinates  of  the  centre  (3,  4),  thus  showing  that  the  degree  of  the 
equation  is  not  changed  by  the  transformation. 

Am.  x'2  +  2/2  +  of  y'  +  6x' 


The  equation  x*  +  y*  =  9  represents  a  circle,  and  the  new  equation 
is  evidently  also  that  of  a  circle,  since  it  is  of  the  form  of  Art.  174. 


PROJECTIONS  OF  LOCI.  \ 


/*>       /' 

Proposition  8.f  O,      ' 

231.  Theorem.— When,  in  any  transformation  of  co-brcU- 
nates,  some  of  the  constants  are  undetermined,  the  new 
axes   may,  in  general,  be  made  to  fulfill   as  many  con- 
ditions as  there  are  independent,  undetermined  constants 
in  the  equations  of  transformation. 

For,  evidently,  definite  values  must  in  some  way  be  assigned  to 
the  undetermined  constants,  in  order  to  fix  the  position  of  the 
new  axes.  One  equation  of  condition  is,  in  effect,  the  determi- 
nation of  the  value  of  one  constant,  two  conditions  of  two  con- 
stants, etc.  (compare  Art.  221). 

232.  Schol. — There   are    evidently   four   independent  constants   in 
the  most  general  transformation  possible;    for  the  axis  of  x'  can  be 
made  to  pass  through  any  two  points,  and  i/  through  any  other  two ; 
or  each  can  be  made  tangent  to  two  different  loci,  etc.,  etc. 


Proposition  #.f 

233.  Theorem.— Replace    x    by    -x',   or    y    by    ~y',   in  any 

a  o 

equation  of  a  locus,  and  the  resulting  locus  will  be  an 
orthogonal  projection  (Art.  60)  of  the  first  locus  upon  some 
oblique  plane. 

For,  evidently,  if  the  axis  of  x 
be  supposed  to  pass  through  0, 
and  to  be  perpendicular  to  the 
paper,  and  the  angle  YOY'  be 

such  that     -=sec    YOY1 ';  then, 

* 
if  the  locus  of   (x,  y)  is  in  the 

plane  XOY,  when  y=°^y',  or  —,  =  %  the  locus  of  (x,  y'},  in  the 
plane  XOY',  will  be  at  the  foot  of  the  perpendicular  upon  XOY' 
through  (x,  y).  The  same  may  be  proved  if  x  =  -xr. 


90 


EQUATIONS  AND  LOCI. 


234.  Schol. — When  p  is  replaced  by  —  in  any  equation  expressed 

in  polar  co-ordinates,  the  scale  of  curve  represented  is  thereby  changed ; 
e.  g.,  if  m  =  2  the  scale  is  doubled,  and  if  m  =  -§•  the  scale  is 
decreased  one  half. 

/i 

When  0  is  replaced  by  — ,  the  loops  or  other  parts  of  the  curve  which 
were  contained  in  any  angle  0t  are  caused  to  be  contained  in  n  times 
that  angle ;  e.g.,  if  n  =  f ,  the  part  of  the  figure  which  was  described 
by  any  single  rotation  of  the  radius  vector  through  360°  will  be  com- 
pletely described  during  the  rotation  of  the  radius  vector  through  180°. 

This  will  appear  more  clearly  in  the  chapter  on  polar  curves  and  spirals. 


Proposition  10. 

235.  Tlieorem. — The  general  equation  of  the  nth  degree 
represents  not  only  the  curves  peculiar  to  that  degree,  but 
also  all  curves  represented  by  equations  of  inferior  degrees. 

For,  since  in  a  general  equation  the  constants  may  have  any 
finite  value,  among  other  values  are  those  which  will  enable  us  to 
separate  the  equation  into  factors,  whose  product  is  of  the  nth 
degree,  and  equal  to  zero.  Since  the  product  =  0,  by  the  "  Gene- 
ral Theory  of  Equations,"  each  factor  =  0,  and  so  represents  a 
curve  of  lower  degree. 


236.  Schol. — The  product  of  two 
equations  of  loci  represents  at  once 
the  two  loci.  E.  G.  Take  the  two 
equations  x  +  y  —  0  and  x  —  y  =  0, 
whose  product  is  x2  —  y1  —  0. 


.  *  .     y  =  dr 

every  value  of  x  we  have  two  values  of  y,  equal,  but  with  opposite 
signs. 

It  may  also  be  noticed  that  the  product  of  two  equations  of  loci  put 
equal  to  some  constants  represents  a  locus  which  approximates  some- 


AXES  OF  SYMMETRY.  91 

what  to  the  two  loci.  -27.  G.  The  equation  xi  —  yl  =  l  represents  a 
curve  approximating  in  position  to  the  lines  x  +  y  =  0,  x  —  y  =  0 
— *.  e.,  x^-yl  =  0  (see  fig.  in  Art.  358). 


Proposition  11.^ 

237.  Theorem.—  <A  locus  has  an  axis  of  symmetry  in  the 
following  cases: 

1st.  It  is  symmetric  about  the  axis  of  x  when  its  equa- 
tion is  unchanged  by  a  cliangc  in  the  sign  of  the  y  co- 
ordinate —  i.  e.,  the  equation  contains  only  even  powers  of  y. 

For,  then,  each  value  of  x  corresponds  to  two  points  (x,  y)  and 
(x,  —  y)  which  are  situated  symmetrically  about  the  axis  of  x. 

E.  G.  y2  =  ax3    (fig.  in  Art.  393). 

2d.  It  is  symmetric  about  the  axis  of  y  when  its  equa- 
tion involves  only  even  powers  of  x. 

E.  G.  xz  =  4py    (%  in  Art.  245). 


3d.  It   is   symmetric   about   both   the  axes  of  x    and    y 
when  its  equation  involves  only  even  powers  of  both  x  and  y. 

E.  G.  x*  +  y2  =  r*    (Art.  171). 


4th.  It  is  symmetric  about  the  bisector  of  the  angle  XOY 
when  its  equation  is  unchanged  by  interchanging  x  and  y. 

For,  to  each  point  (x,  y)  corresponds  a  point  (y,  x)  situated 
symmetrically  about  this  bisector.  E.  G.  xy=m  (Art.  361). 

5th.  It  is  symmetric  about  the  bisector  of  the  angle  XO  Y 
when  its  equation  is  unchanged  by  substituting  for  x  and 
y,  —  y  and  —  x  respectively. 

For,  to  every  point  (x,  y)  corresponds  a  point  (—  y,  —x}  situated 
symmetrically  about  this  bisector.  E.  G.  xy=m. 


92  EQUATIONS  AND  LOCI. 

6th.  It  is  symmetric  about  both  these  bisectors  when  the 
equation  is  unchanged  by  substituting  for  x  and  y,  both 
y  and  x,  and  also  —y  and  —x  respectively. 

KG. 


Proposition  l£.f 

238.  Theorem.—  1st.  A  locus  has  four  parts  or  branches 
alike  when  its  equation  is  unchanged  by  substituting  for 
x  and  y  either  y  and  —x  or  —y  and  x  respectively. 

For,  on  putting  instead  of  x  and  y,  y  and  —  x  respectively, 
every  point  (x,  y}  has  a  corresponding  point  (y,  —re)1  which  bears 
the  same  relation  to  the  origin  and  the  axis  of  y  that  (x,  y)  bears 
to  the  origin  and  the  axis  of  x  —  i.  e.,  one  part  or  branch  of  the 
locus  rotated  90°  in  its  plane  about  the  origin  then  coincides  with 
another  part  or  branch.-  In  the  same  way  make  another  and 
another  rotation,  thus  showing  that  the  curve  is  alike  in  the  four 
angles. 

KG. 


2d.  A  locus  has  two  parts  or  branches  alike  when  its 
equation  is  unchanged  by  substituting  ,  for  x  and  y,  —x 
and  —  y  respectively. 

For  two  applications  of  the  operation  in  the  previous  case  is 
this  operation  —  i.  e.,  a  rotation  through  180°. 

KG.          a*-*  =  L 


CHAPTER   VI. 

THE   PARABOLA. 

INTRODUCTORY   TO  THE   CONIC. 
239.  Definition.— The  equation 


represents  a  conic  section;  hi  which  p  is  the  distance  of 
any  point  of  the  conic  from  a  given  focus,  and  d  is  the 
distance  of  that  -point  froin  a  given  right  line,  called  the 
directrix,  and  e  is  any  constant  called  the  eccentricity. 

The  curve  is  symmetric  about 
the  axis  (see  Art.  5).  Let  the  fixed 
line  be  the  axis  of  y,  and  the  per- 
pendicular to  it  through  F,  the  axis 
of  x;  then,  expressing  the.  above 
equation  in  rectangular  co-ordinates, 


let  OF=8pt  then  p  = 


(x— 


and  d  =  x,  .'. 


(x  —  %pf  =  ex. 


240.  .  '  .  The  equation 


represents  a  conic  referred  to  a   directrix  and  principal 


axis. 


When  e  <  1,  the  conic  is  called  an  ellipse. 
When  e  =  l,  the  conic  is  called  a  parabola. 
When  e>  /,  the  conic  is  called  an  hyperbola. 


93 


94  THE  PARABOLA. 


241.  Exercise.  —  The  equation 


is  the  general  form  of  />  —  ed  in  rectangular®. 

242.  A  Focal  Chord,  or  Parameter,  is  any  chord  of  the  curve 
passing  through  the  focus. 

The  Latus  Rectum,  or  parameter  of  the  principal  axis,  is  parallel 
to  the  directrix. 

A  Diameter  is  the  right  line  which  is  the  locus  of  the  centres 
of  parallel  chords,  as  will  be  shown  hereafter. 

The  Centre  is  the  point  of  intersection  of  diameters,  as  will  be 
shown  hereafter. 

The  Vertex  is  at  A,  in  the  figure  of  Art.  239. 

Conjugate  Diameters  are  so  situated  that  each  bisects  a  system 
of  chords  parallel  to  the  other. 

.  Supplementary  Chords  extend  from  any  point  of  a  curve  to  the 
extremities  of  the  same  diameter. 


THE   PARABOLA. 

Proposition  1. 
243.  Tlieorem.—The  equation 


represents  a  parabola;  in  which  x  and  y  are  the  rect- 
angular co-ordinates  of  any  point  of  the  curve,  and  4p 
is  the  latus  rectum,  when  the  origin  is  at  the  vertex,  and 
the  axis  of  x  is  the  axis  of  the  curve. 


RECTANGULAR   EQUATION. 


95 


For,  if  (Art.  240)  e  =  l, 
then  y*  +  (x—2p)*  =  a?. 

Beducing, 

y2  =  4p(x-p).  ..(a.) 

Let  P  coincide  with  A  ; 
then  y  =  #.  .'.  OA=x=p. 
But  (Art.  239)  if  e  =  l, 
then  f>  =  S.  .'  .  OA=AF-, 
.  '  .  the  vertex  of  a  parabola' 
bisects  the  distance  between 
the  directrix  and  focus. 
Move  the  origin  to  A,  then 

x0=p,  and  yQ  =  0', 
.  '  .     (Art.  23)  x  =  p  +  x', 

and          y=y'' 
Substituting  these  values  of  x  and  y  in  (a.),  we  have, 

y'2  =  4px',  or  y'  =  ±  g 
Omit  the  primes  :     .  *  .     y^ 


244.  Cor.  1.  —  If  the  origin  be  moved  any  distance  x0  =  ±  a  on  the 
axis  of  x,  the  equation  becomes  (Art.  23)  y2  = 


245.  Cor.  £.  —  The  equation  y2  =  ^[pa:  represents  a  parabola  in  the 
position  AOB,  with  the  axis  of  y  tangent  at  the  vertex. 


y*  =  —4px  represents  A^OB^ 
with  axis  of  y  tangent  at  the  vertex. 

x*  =  4py  represents  A^OB^ 
with  the  axis  of  x  tangent  at  the  vertex. 

rr2  =  —  4py  represents  A3OI>3, 
with  the  axis  of  x  tangent  at  the  vertex. 


96 


THE  PARABOLA. 


246.  Cor.  3. — If  in  the  equation  y2  =  4px  we  let  x=p,  then 
y  =  ±:2p  =  j-  (latus  rectum'} :  and  (Art.  243)  the  distance  from  the 
vertex  to  the  focus,  or  from  the  directrix  to  the  vertex,  is  p  =  j-  (latus 
rectum). 

Also,  if  y2  =  4p%,  then  x  :  y  :  :  y  :  4pt  • '  •  in  the  parabola  the 
latus  rectum  is  a  third  proportional  to  the  abscissa  and  ordinate. 

If  (#!,  3/1)  and  (x2,  y2)  are  upon  the   parabola,  then         y*  = 
and          2 


243.  Schol.  1. — Since  p  =  d  =  x'  +p,  omit  the  prime,  and  the  equa- 
tion p  =  z  -\-p  is  called  the  linear*  equation  of  the  parabola.  By  it  the 
parabola  may  be  constructed. 


248.  Schol.  2.—T he  equation 
p  =  d  enables  us  to  construct  the 
parabola  by  continuous  motion,  with 
the  aid  of  a  ruler  CD,  a  triangle 
ABC,  and  a  thread  equal  to  AB 
fastened  at  A  and  F.  If  a  pencil  P 
be  kept  in  contact  with  the  triangle 
along  AB,  and  the  triangle  slide 
along  the  ruler,  evidently  PB  =  PF, 
and  the  locus  of  P  is  a  parabola. 


A 


Proposition  2. 
249.  Tlieorem.—The  equation 


represents  a  line  tangent  to  the  parabola    y*  =  4px 
point    (xi,  ?/i)    of  the  curve. 


*  An  equation  of  the  first  degree  ia  frequently  called  a  linear  cquati 


TANGENT. 


97 


For  (Art.  90), 


**-M 


/j>  \ 


represents  a  right  line 
through  two  points  P2 
and  P3.  If  these  points 
are  upon  the  parabola, 
the  equations  of  con- 
dition which  must  hold 
are  y3=4px3) 

and       y} • 
Subtracting, 


+ 2/3)  =  4p  fe 
-3      4 


.  •".     substituting  in  (6.)  -      -=      _f    .  .  .  .  (c«) 

x     #3     2/2  -t-  2/3 

If  P2  and  P3  coincide  at  PD  then  y2=  y^^=ylt  and  (c.)  becomes, 

rf>~ 


which  is  the  equation  of  the  tangent  line.     From  this 

w/i  -  y?  =  %PX  ~  ®PXI  '>  but  2/i2  =  4pzi,   •  *  •  yyi=  -P  (x 


250.  Cor  1.  —  At    the    point    of    contact    the    equation    becomes 
y\=4pxi,  as  it  should,  since  the  point  Pl  is  on  the  parabola  and  on 
line  also. 

251.  Cor.  2.—  If  y  —  0,  then  2p  (x  +  Xl)  =  0,  .  •  .  x  =  -  x,, 

.'.     the   intercept    a  =  xl;    i.e.,   AO=OAi,  and  AAi,   which  is 
called  the  subtangent,  is  bisected  at  the  vertex. 

Also  (from  ^m.  tri's)  AO  :  OAl  :  :  AB  :  £Plt  .  '  .  AB  =  BPV. 


98  THE  PARABOLA. 


252.  Cor.  3.—  If        y  =  0,      then,  a  =  x. 
.'.     (Art.  247),      Pl=p  +  Xi=p  +  a.        —i.e.,    AF=FPl, 
.  •  .     AFPi  is  an  isosceles  triangle,  and      FAPl  =  AP^F. 
But  if  ED  is  parallel  to  AX,  then  FAP,  =  EP^A. 
.  •  .     the  tangent   A  T  bisects  EP^F,   and   EAFP^  is  a  rhombus 
whose  diagonals  bisect  each  other  at  right  angles  on  the  axis  of  y. 

Again,  from  the  figure,  DPl  T=  EP^A.     .  '  .     AP,F=-  DP±  T. 
If  CPl  is  the  normal  at  Plt  then  AP^=  CP1T=90°. 
Subtract  .-.     AP.C-  AP,F=  CP,T-  DP^T. 


Also  (from  sim.  tri's)  AB  :  BP,  :  :  AF  :  FC, 

.  '  .     (Art.  251)  AF=  FO  =  Pl. 
.  '  .     CFPi  is  an  isosceles  triangle,  and  FP1C=P1CF. 


253.  Cor.  4—  If  x  =  0,  then  b  =  0£  =  y  = 

yi 

But  yl  =  2^/px^  .  •  .  b  =  y'pxlt 

and  AB  =  y'aT+V1  =  yV  +_pa^=  v/^l  (Art.  247). 

Again  a  :  b  :  :  yV  +  V2  :  r,  (=  FB) 

or  a?!  :  -\/px±  :  :  i/p^  :  r.  .'  .  r  — 

Also  from  (sim.  tri's)  it  may  be  shown  that 


and  that  CPl  =  2BF= 

254.  Cor.  5.—  From  the  figure 

%p  i 

—  =  tan     =  tan  r. 

y\         x 

T,    ,  .  tan2r 

By  trig.  sin2  r  —  -  -  , 

J 


TANGENT.  99 


^^TTV^ 

y? 

But  =      *  .'.     sin2r  = 


But 


sm2  r 


a  form  of  the  linear  equation  of  the  parabola  of  some  importance  ;  in 
which  pl  is  the  radius  vector  of  P,,  and  r  the  angle  between  the  tan- 
gent at  PI  and  the  axis  of  x. 

Also  since  -±-  =  tan  r  ; 


sin  T 
=  -  ;     .  '  .     2  p  cos  r  —  y,  sin  r  =  0. 

cos  r 


255.  Examples.  —  Construct  the  tangents  represented  by  the  follow- 
ing equations,  with  the  parabolas  to  which  they  are  tangent,  finding 
the  equations  of  the  parabolas,  when  yl  —  4- 

(1.)        y 

(2.)        y 

(3.)  Find  the   equation  of  the  tangent  to  the  parabola  y*  =  1 
when  ^  =  9.  Ans.  3y  =  2x  +  18. 


Proposition  5.f 
256.  T1ieorem.—The  equation 

P 

y  ~  mx  +  — 

also  represents  a  line  tangent  to  the  parabola   y*  —  4p%\    v 
which  m  =  —  =  tan    . 

y\ 


100 


THE  PARABOLA. 


i)  and  y?  = 


For,  if  (Art.  249)  yyl  = 
then,  by  substitution,      yyl  = 

•'•     y  = 

This  is  called  the  magic  equation  of  the  tangent  line. 


,  or  #!  =  — , 

4p 


y  =  m#  + 


m 


257.  Schol.  1. — The  locus  of  the  foot  of  the  perpendicular  from  the 
focus  upon  the  tangent  line  is  the  tangent  at  the  vertex.     For,  the 
equation  of  the  tangent  may  be  written  my  —  m'2x  =p,  and  the  per- 
pendicular to  it  through  the  focus, 

(p,  0),  is  (Art.  128)       my  +  x  —p. 

Subtract,      .  • .     (1  +  m2)  x  =  0 

.  • .  x  =  0,  which  is  the  equation 
of  the  tangent  at  the  vertex. 

This  enables  us  to  construct  a  pa- 
rabola approximately  by  drawing 
successive  tangents,  as  in  the  figure. 

258.  Schol.  2.— The  locus  of  the 
intersections    of    tangents    perpen- 
dicular to  each  other  is  the  directrix. 

P. 


For, 


=  mz  -f 


—  1 


represents  some  tangent ;  in  which  equation  let  us  replace  m  by  - 


(Art,  123), 


y  =  — mp   represents  a  second  tangent  per- 


m 


pendicular  to  the  first.     Subtract,  etc., 
equation  of  the  directrix. 


x  =  — p,  which  is  the 


259.  Exercise. — Prove  that  the  equation  y  =  m<&  +  —  is  the  locus 

ra0 

of  the  intersection  of  the  tangent  with  a  line  drawn  from  the  focus, 
and  making  with  the  tangent  line  an  angle  whose  tangent  =  m0. 


POLE  AND  POLAR. 


101 


Proposition  4. 

260.  Theorem.— The  equation 


also  represents  a  right  line  which  is  the  chord  of  contact 
of  two  tangents  drawn  to  the  parabola  y.2  =  4px  from  the 
external  point  (aplf  y^. 

For,  let  P2P3  be  the  chord  of  contact  of  the  tangents  PtP2 
and  P,PS. 

^— ^ .  .  .  (d.)      represents  P2P3. 

JO       JC<*        %Cn         fcCo 


From  Art.  90, 


From  Art.  249,  yy2  =  2p  (x  +  x2)  ..  (e.) 
represents  the  tangent  at  P2,  and  if 
it  passes  through  Pv  the  co-ordinates 
of  PI  must  satisfy  (e.). 


=  2p  (xl  +  x3)  .  .  .  for.) 


-  =  — ,     substitute  in  (d.).     .  • . 


similarly 

subtract  .-. 


clear  of  fractions.      .  * .     yyl  —  y{yz  =  %px 

Add  (0r.).     .  * .     yyl  =  2p  (x  +  x^  represents  P2P3. 

PI  is  called  the  pole,  and  P2P3  its  polar,  with  respect  to  the  parabola. 

261.  Exercise. — If  the  pole  is  on  the  directrix,  show  that  the  polar 
is  a  focal  chord. 

Proposition  5. 

262.  Theorem— The  equation 


also  represents  a  right  line  which  is  the  locus  of  the  inter- 
sections of  the  pairs  of  tangents  to  the  parabola  ?/  =  4px, ' 
drawn  from  the  extremities  of  all  chords  which  pass  through 
any  fixed  point  (xit  yi). 


102 


THE  PARABOLA. 


For,  let  P!  be  the  fixed  point  through  which  all  the  chords 
pass,  and  let  QiQ2  be  any  chord  through  Pr     If  the  tangents  at 


Ql  and  Q2  meet  in  some  point  P2,  then  (Art.  260)  yyz  =  2p  ( 

is  the   equation  of  the  chord   §1%;    an(^  at  -^i  this   equation 

becomes  y\yz  =  ^P  (x\  +  #2)  •  •  •  ('*•) 

Similarly,  if  the  tangents  at  the  extremity  of  anothejr  chord 
through  Pl  intersect  at  P3,  y^  =  gp  (xl  +  x3)  .  .  .  (fa) 


But  (Art.  90), 


,  --±~^f  -  •  -  (<•)  represents  P2P3. 


x-x 


Eliminate  as  in  Art.  260.    .  * .   yyl=%p(x  +  xj  represents  P2P3. 

Pi  is  called  the  pole,  and  P2Ps  its  polar,  with  respect  to  the  parabola. 


263.  ScJtoL—  The  tangent  (Art.  249)  yyi  =  2p(x  +  xl)  is  the  par- 
ticular case  in  which  the  pole  is  upon  the  curve,  and  consequently  upon 
its  own  polar. 

264.  Examples.  —  (1.)  Given  a  parabola  whose  latus  rectum  is  8, 
to  find  the  polar  of  the  point  (#,  7).  Ans.  7y  =  4x+  12. 

(2.)  The  latus  rectum  of  a  parabola  is  4  1  find  the  pole  of  the  line 


(3.)  Given  a  parabola 
cepts  of  its  polar. 


z,  and  a  point  (—  4,  10),  to  find  the  inter- 
Ans. a  =  4,  b  =  —  ~. 


NORMAL  LINE.        SUBNORMAL.  103 

265.  Exercise. — If  the  focus  is  the  pole,  the  directrix  is  the  polar. 

Proposition  6. 

266.  Theorem.— The  equation 

y-yi  =  -;z(*-*i) 

vp 

is  that  of  a  line  normal  to  the  parabola    yi~4px    at  the 
point    (xl}  ?/i). 

For,  resuming  the  equation  of  the   tangent  line   (Art.  249), 


J         jl          „.      \  I/) 

yi 
we  have  (Art.  123),        y~y\  =  ~~^~  (%  —  #i)» 

perpendicular  to  the  tangent,  and  it  passes  through  the  point 
of  contact  (xl}  yx)  by  Art.  98. 

267.  Schol.  1. — The   equation  y  —  yl  —  —^-  (#  —  #  )  also  repre- 

2p 

sents  the  normal  when  the  origin  is  at  any  point  on  the  axis  of  x. 

Move  the  origin  so  that  the  axis  of  y  shall  pass  through  the  point 

?/         oc 
(#i>  y\) — i.  e.,letxi  =  0;  then  _ -f  —  =  1,   and   (figure   of  Art.   249) 

y\     %p 

AiC=%p        . ' .     subnormal  =  a  constant. 

268.  Schol.  £.f— Let  m=    -  y-  =  tan  ^,         .  • .    yl  =  —  2pm. 

Also,  since    y\=4px\,      •"•     Xi=pl  —  \=pmz. 

Substitute  these  values  in    y  —  yl  =  —  —  (x  —  #j), 

then,        y  +  2pm  =  m(x  — prri?) 
or,  y  =  mx  — pm  (2  +  w2), 

which  is  the  magic  equation  of  the  normal  to  the  parabola. 


104 


THE  PARABOLA. 


269.  Examples. — (1.)  Find  the  equation  of  the  normal  to  the  parab- 
ola, when  p  =  2  and  x±  =  32.  Ans.  y  =  —  J^x  -f  144- 

(2.)  Find  the  equation  of  the  normal  to  the  same  parabola,  passing 
through  the  point  (10,  4)  not  on  the  curve,  the  equation  of  condition 
that  the  normal  shall  pass  through  a  point  (x.2,  y2)  being  (Art.  266) 


y\— 


y\ 


Am. 


xl  — 


=  24  is  normal  at  (8,  8). 


Proposition  7. 

270.  Theorem.  —  The  equation 


represents  a\  parabola  inferred  to  a  tangent,  and  a  parallel 
to  the  axis  through  the  point  of  contact;  in  ivhich  p'  is 
the  distance  from  the  focus  to  the  point  of  contact,  and 
4pf  is  the  length  of  the  focal  chord  parallel  to  the  tangent. 

For,  transfer  the  origin 
from  the  vertex  0  to  some 
point  Of  upon  the  parabola 
y2  =  4px.  .  .  .  (I.)  The  equa- 
tions of  transformation  are 
(Art.  89,  3), 


i 


Mw 

all 


0  /F 


x  =  xl  +  xf  +  y'  cos  T 


when  we  make  the  axis  of  xf  parallel  to  the  axis  of  x,  and  the 
axis  of  y'  the  tangent  at  0'.  Therefore,  substituting  these 
in  (I.), 


(yl  +  yf  sin  r)2  =  4p  (x1  +  xr  +yr  cos  r), 
or  expanding  and  arranging, 

'  (#p  cos  r  —  yl  sin  r). 


y'2  sin2 


TANGENT  AND  DIAMETER  AS  AXES.  105 

Bat,  since  (xlt  yt)  is  on  the  curve,  y?  —  Jt,pxl  =  0, 
and  (Art.  254),  %p  cos  r  —  yl  sin  r  =  0, 

hence          y'2  =  -^-  *'  ;         .  •  .     (Art.  254),  y'2  =  ^  x'. 

bill       4 

Again  (Art.  252),  f>l=AF=  O'C. 

Let    x'=Pl  =  0'C.      .'.    y'2  =  4(>l2.       .'.    y'  =  ±2Pl  =  CQ. 

If,  for  convenience,    pl=p',    then,   4pi=4pf   is  the-  length  of 
the  focal  chord  parallel  to  the  axis  of  y'. 
Hence,  omitting  the  primes,  y2  =  4pfx. 


271.  Cor.  —  Since  y=±  #i/f&,  every  positive  value  of  x  gives  two 
equal  values  of  y,  or  the  axis  of  x  is  the  bisector  of  all  chords  parallel 
to  the  tangent,  and  is  therefore  a  diameter,  and  all  lines  parallel  to  the 
principal  axis  are  diameters. 

272.  Schol.  —  Since    parallel    chords    all  pass    through    the    same 
point  (oo!,  oo2)  at  infinity,  and  since  for  this  point  x\  =  oc^  and  y^  =  oo2, 
the  equation  of  the  polar  of  this  point  (Art.  262)  yy^  =  2p  (x  +  xt) 


v   ,  i  .. 

or    v=    --  r  —  —    becomes  y  —  —  —  ,  or  y  =  a  constant.     I  his  polar 

y\      2/1  3/1 

of  parallel  chords  is  then  (Art.  106)  parallel  to  x,  and  cuts  parabola  at 
the  point  of  contact  of  the  tangent  parallel  to  the  chords.  Hence  the 
locus  of  the  intersections  of  pairs  of  tangents  at  the  extremities  of 
parallel  chords  is  the  diameter  bisecting  those  chords. 


Proposition  8. 

273.  Theorem.— The  equation 


represents  a  parabola;  in  which  p  is  the  radius  vector, 
and  0  the  variable  angle  of  any  point  of  the  curve,  mea- 
sured from  the  vertex,  when  the  pole  is  at  the  focus,  and 
l  =  2  is  the  semi-latus  rectum. 


106 


THE  PARABOLA. 


For,  y2=4p(%+p)  is  ^e 
equation  of  a  parabola  referred 
to  the  rectangular  axes  through 
the  focus.  Transform  by  the 
equations 

x  =  —  p  cos  6  and  y  =  p  sin  6. 


Cancel 


V 

±l-cos6)     &p(±l  —  cos  0) 
~7in20  l-cos?d       ' 


2p  ,  £p 

P  =  TT    71i>    and    ~P  =  T^ 


1  +  cos  d' 


1  —  cos  6 


which  are  the  positive  and  negative  values  of  p  corresponding  to 
the  angle  6. 

274.  SchoL—By  trig, 

/?  /J 

1  +  cos  0  =  2  cos2  -,   and  1  — cos  0  =  2  sin2  -. 

2  2 


COS0 


0' 


COS 


2  __ 


275.  Exercise. — Prove  that 


cos 


is  the  polar  equation  of  the  parabola  when  the  pole  is  at  the  vertex. 

Proposition  9.f 
276.  Theorem.— The  equation 


represents  a  line  tangent  to  the  parabola    p  = at 

1  +  cos  0 

the  point    (j>lt  0J,    the  pole  being  at  the  focus. 


POLAR  EQUATION  OF  TANGENT  LINE.  107 

For,  yyl  =  £p(x+xl  +  2p)  is  the  equation  of  the  tangent  line 
referred  to  rectangular  axes  through  the  focus.  Transform  by 
equations 

x  =  — j0cos#,   and  y=psmO', 

.  • .     ppl  sin  6  sin  Ol  =  —2pp  cos  6  —  £p  pl  cos  0l  +  4p2. 


'l  +  COaOj? 

Substitute,  clear  of  fractions,  transpose  and  solve  for  p, 


r     cos  (6  -  0J  +  cos  6     cos  (6  -  0L)  +  cos  6' 

277.  ScJiol.  1. — If  0  =  01}  this  becomes  the  equation  of  the  curve,  as 
it  should. 

278.  Schol.  2. — By  trigonometry; 

I  P 


cos2  (d  —  0!)  +  cos  0 


279.  Examples. — (1.)    Construct    the    tangent    to    the    parabola 
p  = 1 at  the  point  whose  vectorial  angle  is  Ol  =  60°,  and  find 

7  +  COS0 

the  angle  which  it  makes  with  the  initial  line. 

.Ans.  The  angle  =  60°.  • 

(2.)  Find  from  (Art.  280)  the  normal  to  the  same  parabola  at  the 
same  point. 

280.  Exercise. — Prove  that  the  equation  of  the  normal,  when  the 
pole  is  at  the  focus,  is 

*«4 


If  0  =  0l}  this  also  becomes  the  equation  of  the  curve. 


CHAPTER    VII. 
HYPERBOLA  AND   ELLIPSE. 

Proposition  1. 
281.  Theorem.— The  equation 


—  7>2 


represents  an  hyperbola  with  the  origin  at  the  centre;  in 
which  a2  and  —  b*  are  the  squares  of  the  intercepts  on 
x  and  y  respectively—  i.  e.,  the  semi-axes. 

For,  y 


x—8p)*  =  #tf.  .  .  (a.)   is  by  definition   (Art.  240) 


OA 


the  equation  of  an  hyperbola  when  e  >  1,  with  the  origin  at  0. 
If  y  =  0,     then     x  —  2p  =  d-  ex, 


and  .  * . 

is  the  intercept  OA2. 

108 


=  Sp_ 
1  +  e 


intercept  OAl}  and    XQ  =  73 

JL  C/ 


RECTANGULAR  EQUATION  OF  THE  HYPERBOLA.         109 
Let  CA^  =  +  a,     and     CA2  =  —  a, 

then  (Art,  9),  CAt  =  J-  (OAt  -  0^2)  =         -          =  -  =  a. 


Again,  07=1 


Move  origin  to  C  (Art.  23),     .  •  .     y=y',  and  x  --=xr  ---  , 

6 

and  substituting  these  in  (a.), 


...     y'2  =  a2-  x'2  -  e2  (a2  -  x'2)  =(1-  e2}  (a2  -  x'2). 
If  x'  =  0,  then  y/2  =  a2(l-  e2), 

the  second  member  of  which  is  negative  when  e>l, 
.'.  'let         a*  (1—  #)=-&; 

then,          T/'2  =^jr(a2-  x'2},  or  ~  +  ^  =  1,  .  .  .  (b.) 

(dropping  the  primes)  represents  the  same  curve  that  (a.)  do^s. 

282.  Cor.  1.  —  This  curve  is  in  form  symmetrical  about  both  the 
axes  of  x  and  y,  for  every  value  of  x  gives  two  values  of  y  numerically 
equal,  but  of  opposite  signs,  and  vice  versa. 

There  are  no  real  values  of  y  in  this  curve  between  x  —  +  a  and 
.>;  =  —  a,  as  may  be  seen  from  the  equation,  when  put  in  the  form, 


110  HYPERBOLA  AND  ELLIPSE. 


07.2 

283.  Cor.  2.—  The  latus  rectum  ED  =  - 

a 

For,  if 


e  e 

let       xf  =  ae;     .-.    y;  =  ^(a*-aV)  =  4.     .-.    Kyt 

(•Id 

284.  Cor.  3.—  Since     a1  (1  -  <*7  =  -  S!,  or  a*  +  V  =  oV, 


for,  (Art.  283)  CF±  =  ae  =  i/a?  +b\  and  from  the  fig.  A&  =  j/a2  +  b\ 

A  1  /™         &  C? 

Also,         C0  =  -  — — . 

285.  Schol.  1. — The  axis  A^  =  2a  is  called  the  transverse  axis. 
The  axis  B^B2  =  2b  at  right  angles  to  the  transverse  axis  is  the  conju- 
gate axis. 

286.  Schol.  2. — The  hyperbola  becomes  equilateral  or  rectangular 
when  the  axes  are  of  equal  magnitude ; 

X*  —  y2  =  a2  is  the  equation  of  an  equilateral  hyperbola. 

287.  Examples.— Find  the  value  of  CFlt  CAl}  CB^  and  ED  for 
the  hyperbola  represented  by  each  of  the  following  equations,  and  con- 
struct their  directrices,  vertices,  foci  and  the  extremities  of  each  latus 
rectum. 

(1.)  £L-| %-  =  !.  Ans.a=±3,  b  =  ±  %/  —  1. 

9        — 4 


(2.)  +      _  =:  1,  Am.  a  =  ±l,b  =  ±  S^l. 

9        — 9 

(3.)  ^-^  =  jr-  Am.a=±2^b  =  ±V—l. 

(4.)  Show  that  the  latus  rectum  is  a  third  proportional  to  the  axes 
2a  and  2b. 

(5.)  Show  that  a  is  mean  proportional  between  CFl  and  CO. 


RECTANGULAR  EQUATION  OF  THE  ELLIPSE. 


Ill 


Proposition  2. 

288.  Theorem.— The  equation 


represents  an  ellipse  with  the  origin  at  the  centre;  in  which 
d*    and    6*    are  the  squares  of  the  semi-axes,  and    a  >  6. 

The  demonstration  of  this 
proposition  is  identical  with 
the  preceding,  with  this  ex- 
ception, that 


since  (Art.  240)       e  <  1. 

The  lettering  of  Art.  281 
will   be   found   to    apply   to 
this  figure,  and  the  results  will  therefore  agree  with  those  in 
Art.  281  on  changing  the  sign  of  b2. 

x2      v2 


289.  Cor.  1. — The  curve  is  symmetrical  ahout  the  axes  of  x  and  y. 

The   values   of  y  are   all  real  between  x  =  +  a,   and  x  =  —  a,  but 

I 

imaginary  beyond,  since     y  =  ±  -\/a2  —  x'\ 

#7,2 

290.  Cor.  2.— The  latus  rectum  = (Art,  283). 


291.  Cor.  3—  Since  a2  (1  -  e2)  =  b\     .  • .     e  = 


lC=ae  =  i/tf^F,     and 


292.  ScftoZ.  Jf.  —  The  axis  ^^  =  ^0  is  the  transverse  or  major 
axis,  and  ^  J52  =  #&  is  the  conjugate  or  minor  axis. 


112  HYPERBOLA  AND  ELLIPSE. 

293.  Schol.  2. — The   ellipse   becomes   a   circle   when   a2  =  b2  =  r2. 
Then  e  =  0,  and  the  equation  becomes  x2  -f  y2  =  r2. 

294.  Schol.  3.— When  5  >  a,  then  e1  =  V/al"y.  is  imaginary,  but 

CL 

e2  =  —        -  is  real,     .  • .     b  is  the  major  and  a  is  the  minor  axis. 
6 


295.  Schol.  4.  —  The  equations  of  the  ellipse  and  hyperbola  may  be 
written  together, 


in  which  the  sign  +  is  for  the  ellipse  and  —  for  the  hyperbola. 

296.  Schol.  5.—  The  equation   2±£ii  _j_  0/  —  .Vi)  =  i   represents 

a2  ±  b'2 

an  ellipse  or  hyperbola  whose  axes  of  reference  are  parallel  to  its  axes 
of  figure  ;  in  which  x^  and  yl  are  the  co-ordinates  of  the  centre.  This 
may  be  shown  by  moving  the  origin  (Art.  23)  in  the  equation 

x2         i/2 

—  -|  —  '-^—^  =  1.     If  Xi  =  zh  «,  then  the  origin  is  at  Al  or  A2,  and  we 

/y.2    —  —    Ortnr-  njl 

have  for  the  equation  at  the  vertex  --  -  ---  h  -~3  =  ^' 

~~ 


,  or  yz  =  ±  —  x^  —  x\ 


297.  Schol.  6. — The     equation     of     the     ellipse    is     (Art.    289), 

-yt  =  i/a2  —  x2,  and  that  of  the  circle  (Art.  171),     yc  =  i/a2  —  x*. 
b 
For  any  value  of  x  common  to  the  ellipse  and  the  circle  we  have, 

.  • .     yc  =  -  ye,  or  —  =  -.     . ' .     by  Art.  233  the  ellipse  is  a  pro- 

&  V*      ° 

jection  of  the  circle  when  the  major  axis  is  the  line  common  to  both. 

The  same  may  be  proved  of  the  circle  on  its  minor  axis — i.  e.,  for  any 

xc      b 

value  of  ?/  common  to  both,  —  =  — . 

x*     a 


CONJUGATE  HYPERBOLA.  113 

298.  Examples* — (1.)  Change  the  sign  of  the  term  containing  y1 
in  each  of  the  equations  of  Art.  287  from  —  to  -f ,  and  find  the  position 
of  the  foci,  etc.,  of  the  ellipse  represented  by  each  equation. 

(2.)  Show  that  the  proportions  stated  in  Art.  287,  (4)  and  (5),  are 
true  in  case  of  the  ellipse. 


Proposition  3. 

299.  Theorem.—  The  equation 


-a 


represents  an  hyperbola  conjugate  to 

a?        y2 

—  -4-  —  --  / 

a2      -62~ 

—i.  e.,  the  transverse  axis  of  the  one  is  the  conjugate  axis 
of  the  other,  and  vice  versa. 

For,  evidently  this  represents  the  curve  whose  vertices  are  at 
B^  and  JB2  (Art.  281),  and  it  may  be  considered  to  be  derived  from 

x2       y2 
--  h  —  —  =  / 
a2  *  -b2 

x  11 

by  the  exchange  of  -  and  ±  7.     Also  since  —  a2  is  the  square 

C6  0 

of  a  semi-axis,  that  axis  is  imaginary,  while  the  other  is  real. 

Similar  corollaries  and  scholia  apply  to  this  proposition  and  Proposition  1. 
300.  Cor.  —  In  the  conjugate  hyperbola 


for,  CFt  =  A&  =  CFl  =  i/a»  +  b\  and  CBl  =  b. 

Also,  the  distance  from  0  to  the  directrix  =  -  =         2 — — 


s 


114  HYPERBOLA  AND  ELLIPSE. 

Proposition  4* 
301.  TJieorem.—The  equation 

• :  -  a?          y2 

—  +-Z—=1 
—a2       —  b2 

represents  an  imaginary  ellipse. 


For,  from  this  equation  y  =  ±  -  y  —  a?  —  x2,  hence  all  values  of 
x  give  imaginary  values  of  y  ;  still  either  el  or  e2  is  real,  and  has 
the  same  value  as  in  the  real  ellipse  (Art.  294). 

v  -Proposition  £. 
302.  Theorem.—  The  equations 

^+y--1         nd        -—        ~^—        1 

a2^b2  a2±d2~i~b2±d2~ 

represent  conies  having  the  same  foci,  and  are  called  con- 
focal  conies. 

For,  in  the  first  curve  (Art.  288)  a2e2  =  a2  —  b2,  and  in  the  second 
(a2  ±  d2)e2  =  a2±d2-  (b2  ±  d2)  =  a2  -  b2  ; 

.  *  .  the  distance  from  the  centre  to  the  focus  in  any  conic 
represented  by  the  equation 

x2  .  y2 


is  j/a2  —  b2,  whatever  value  d2  may  have. 
This  is  true  whether  b  is  real  or  imaginary. 

303.  Examples.  —  (1.)  Find  the  nature  of  the  curves  represented 
by  the  equation  (d.)  when  a2  =  4,  b'2  =  1,  and  d2  =  successively  2,  1, 
0,  with  the  upper  sign,  and  1,  2,  3,  4,  with  the  lower  sign. 

(2.)  Find  the  vertices,  etc.,  of  the  hyperbolas  conjugate  to  those  of 
Art.  287. 


LINEAR  EQUATION  OF  THE  HYPERBOLA. 


115 


, 


Proposition  6. 
304:.  Theorem.— The  equations 

pl  =  ex  —  a    and    p2  = 
also  represent  the  hyperbola 

x*          v2 


in  which    pi    and    pz    are  the  focal  radii  vectores. 

For,  since  (Art.  239)  p  =  ed  and  S  =  x  ±  CO  =  x  ±  -, 
when  the  origin  is  at 
C,  the  sign  +  or  - 
being    used,   accord- 
ing as  the  focus  and 
directrix  at  the  left 
or  right  of  C  is  em- 
ployed   to     describe 
the  hyperbola, 


=  e 


/        a\ 

I  x  ±  -  1    .  *  .     pi  =  ex  —  a,     and    p2  =  ex  +  a. 
\        e/ 


305.  Schol.  —  Sub- 
tracting, /?2  —  PI=  %a, 
is  the  equation  of  the 
hyperbola  in  focal  co- 
ordinates. This  equa- 
tion enables  us  to  con- 
struct the  hyperbola 
by  continuous  motion 
as  follows :  in  a  piece 

of  thread  many  times  longer  than  %a  make  a  loop  so  that  its  knot 
shall  divide  the  thread  into  two  segments  whose  difference  is  2a,  and 
fix  its  extremities  at  f\  and  F*  so  that  I\J?2  =  2ae.  Then,  if  both  seg- 
ments of  the  thread  slide  in  a  notch  near  the  point  of  a  pencil  at  P,  the 
.loop  L  being  held  in  any  direction  such  as  to  keep  both  parts  of  the 


116 


HYPERBOLA  AND  ELLIPSE. 


string  taut,  and  paid  out  as  the  pencil  advances,  then  the  hyperbola  is 
described.  For,  F^PL  is  one  segment,  and  F^PL  the  other,  and  by  con- 
struction F2PL  —  FlPL  =  2a,  .' .  F2P  —  FlP=2a,  or  p2  —  pl  =  2a. 


Proposition  7. 
306.  Theorem.— The  equations 

pl  —  a  +  ex    and    p2=a  — 
also  represent  the  ellipse 


or 


a 


For,  as  in  Art.  304  p  =  ed  and  d  =  -  ±  x,     .  * .     p  =  a±ex. 

307.  Schol. — Adding,  pl  +  p2  =  2a 
is  the  equation  of  the  ellipse  in  focal 
co-ordinates.  To  construct  the  ellipse 
by  continuous  motion,  tie  the  ends  of 
a  thread  whose  length  is  2  ( a  +  -], 

and  putting  it  over  pins  at  F\  and  F2,  let  the  thread  run  in  a  notch  at 
the  point  of  a  pencil  held  against  it,  as  at  P,  the  pencil  will  by  its 
motion  describe  the  ellipse.  For,  the  length  of  the  thread 


-,  and 


=  2a,  or 


Proposition  8. 
308.  TJieorem.—The  equation 

x+y  =  i/az  +  b2 

represents  a  line  passing  through  the  focus  of  an  hyperbola 
and  the  focus  of  its  conjugate. 
Also  the  equations 


and    y  — 


represent  the  directrices  of  an  hyperbola  and  its  conjugate. 


EQUATION  OF  DIRECTRICES,  ETC. 


117 


For  (Art.  284), 
Also  (Art.  300), 

.-.     (Art.  95), 


A"=  CF,  =  ae,  =  -/ 
&  =  CEl  =  be2  =  i/o 


+ 


(+ 


=  1  represents 


CL 


Again  (Arts.  281,  106),  x  =  —  is  the  directrix 


3?  ifi 

irectrix  of  —  +  —    =  1 


d 


.  • .     (Art.  284),  x  = 


and  similarly,  y  =  —  is  the  directrix  of 


CL  0 


+  TJ  =  1, 


309.  Schol. — Add  these  three  equations  with  the  signs  of  the  first 
changed,  and  they  vanish  identically ;  .  * .  (Art.  116),  the  directrices 
intersect  upon  the  line  FlEl  at  J^,  and  the  parallelogram  of  directrices 
is  inscribed  in  the  square  F^E^F^E^.  These  relations  enable  us  to 
construct  the  vertices,  foci  and  directrices  of  an  hyperbola  and  its  con- 
jugate ;  for  aei  =  b&z  =  CFl  —  CE^  =  A^B^ 

and          CO,:  CAL  :  :  CAl  :  OF,     -i.e.,-:a::a:ael 

give  all  the  relations  necessary. 


118 


HYPERBOLA  AND  ELLIPSE. 


310.  Example. — Construct  the  foci,  directrices  and  vertices  of  an 
hyperbola  and  its  conjugate,  when  a2  +  b2  =  18  and  ab  =  6,  a  and  b 
being  the  transverse  axes  of  the  hyperbola  and  its  conjugate  respectively. 


Proposition  9. 
311.  Theorem.—  The  equation 

IE*  ,  _M_  =  / 

a2       ±  62  ~ 

represents  a  right  line  tangent  to  the  ellipse  or  hyperbola 


in  which   Xi    and   2/1    w&  the  co-ordinates  of  the  point  of 
tangency,  and  the  origin  is  at  the  centre. 


A    X 


For,  the  equation 


SO 


is  (Art.  90)  that  of  a  line  through  P2  and  P3.     If  these  points 
are  upon  the  curve,  then  (Art.  295), 

r  2  , ,  2  „  2  ?/  2 

2  j  2  -v  i  3  .^3  •/ 

are  the  equations  of  condition  which  subsist.     Subtract, 


TANGENT. 


119 


Substituting  this  in  the  equation  of  the  line  through  P2  and  P3 


Now  let 
then 


=y2=yi>  and  35  =  3,  =  ^, 


.  • .     clearing  and  transposing,  etc., 


\  , 

2 


_ 

a2       ±6,      a2      =b& 


312.  Cor.  1. — At  the  point  of  contact  the  equation  both  of  the 
curve  and  of  its  tangent  reduces  to 


313.  Cor.  2.  —  If   y  —  0,    the    intercept   of   the    tangent   on  x  is 

Cu  GL  CL    —  OC 

=  XQ  =  —  .   The  subtangent  in  the  ellipse  is  AB  —  --  xl  =  —       —  , 


^     .  •  .     by  Art.  184, 


and  in  the  hyperbola  AB  =  Xi = 

Xi  Xi 

these  subtangents  are  numerically  equal  to  that  of  the  circle  in  which 
r  =  a.     This  enables  us  to  construct  a  tangent  at  (rr1(  yO,  as  follows. 


For  the  ellipse,  draw  from  P2  a  tangent  P2J/1,  and  join  MI  and  PI  ; 
MiPi  is  the  tangent  at  J\.  For  the  hyperbola  draw  M2P2  from  M2 
tangent  at  P2,  and  join  MI  and  PI  ;  MtPi  is  the  tangent  at  PI. 


120  HYPERBOLA  AND  ELLIPSE. 

314.  Example. — Find  the  equation  of  the  tangents  to  the  curves 

x2         y2 

1 — - —  =  1  at  the  points  whose  abscissas  are  respectively  2  and  6. 


315.  Exercise. — Prove  that  the  equation 

xx!  ±  a(x  +  #0  ^  y?/i    = 

is  that  of  the  tangent  line  when  the  origin  is  at  the  vertex. 

Proposition  W.'f 

316.  Theorem.— The  equation 


y  =  mx  ±  j/a2m2  ±  b2 
also  represents  a  line  tangent  to  the  conic 

x2        f-  t 

~  +  ^TTT  =  1 ',   in  which  m  =  tan  x- 

xx        i/i/ 
For,  the  equation         -^  +  «^  =  1  (Art.  311), 

zhZ)2^       .   ±b2 
solved  with  reference  to  yt  is        y  — 1 —  »"i      —  .  .  .  .  (e.) ; 

ay\        y\ 

,          — f—  7)2  x  x 2        11 2 

.  ' .   (Art.  104)      tan^  =  —  -— i — L  =m.  and  since  —^  +  T^TT  =1, 
%  a2  a2       ±62 


Substitute   this   in  (e.),      .'.      y  =  mx  +  j/a2m2  zb  b2}     which   is 
called  the  magic  equation  of  the  tangent  line. 


THE  CIRCLES 


2,  AND 


121 


317.  Schol.  1. — Every  equation  of  this  form  is  tangent  to  the  locus 


318.  Schol.  2.  —  The  equations         y  —  mx  =  -|/a'W  ±  b'2 

x          r~tf 

and    y  =  ---  h  \l  --  ±  &2, 
?tt      \  w2 


or 


represent  two  tangents  perpendicular  (Art.  123)  to  each  other.  Squar- 
ing, adding  and  dividing  by  (1  +  m2),  we  obtain  x2  +  y*  —  a2  ±  &2, 
which  is  the  equation  of  the  circle  which  is  the  locus  of  the  intersection 
of  all  tangents  perpendicular  to  each  other  (cf.  Art.  258). 

319.  Schol.  3.— The  equation    y  = 

or  my  +  x  =  i/V  q=  b'2,  represents  a  line  through  the  focus  (Art.  98) 
perpendicular  to  the  tangent  y  =  mx  +  j/a'2m2  ±  62,  the  co-ordinates 
of  the  focus  being  y\  =  0  and  xl  =  ae  =  -j/a2  +  b'2.  Squaring,  adding 
and  dividing  by  (1  -f  m2),  we  find  x2  +  y*  =±  a2 ;  hence  the  locus  of  the 
foot  of  the  focal  perpendicular  upon  the  tangent  is  a  circle  whose 
radius  is  a. 


This  principle  affords  a  construction  of  the  ellipse  and  hyperbola 
similar  to  that  of  the  parabola  in  Art.  257.  Draw  a  circle  upon  the 
transverse  axis,  and  through  either  focus  draw  chords  to  the  circle ;  at 
the  extremities  of  these  chords  draw  perpendiculars  to  them,  and  they 
will  be  tangent  to  the  ellipse  or  hyperbola.  By  drawing  a  sufficient 
•  number  the  curve  may  be  defined  with  considerable  accuracy. 


122 


HYPERBOLA  AND   ELLIPSE. 


320.  Exercises. — (1.)  Prove  that  the  magic  equation  of  the  tangent 
line  is  y  =  m(x  d=  a)  ±  •\/dimi  +  b'*  when  the  origin  is  at  the  vertex. 

(2.)  Find  the  length  of  the  focal  perpendiculars  p^  and  p2  on  the 
tangent,  and  show  that  pip2  —  ±  b\ 


Proposition  11. 
321.  Theorem.— The  equation 

xxi  ,  yyi 


a        ±: 


represents  a  right  line  which  is  the  chord  of  contact 
of  two  tangents  drawn  to  the  conic 


d2  '  ±b2 

from  the  external  point    (x1(  yL). 


For,  let  P2P3  be  the  chord  of  contact  of  the  tangents  P^  and 


From  Art.  90, 


represents 


From  Art.  311, 


=  -^  ....  (/•) 


POLE  AND  POLAR.  123 


represents  the  tangent  at  P2 ;  and  if  it  passes  through  Plt 

x*x, 
then  - 


Similarly,  — 2"  +  ijT/l  =  ^  •  •  •  • 

Subtract  (/*.)  from  (p.),     .  ' .      ~ — ~2 — ~  +" 


xx,  ,       X.X 


a        ± 


1  3 

.       from  (e).     —    -= 
a;-^3 

,y»  xx,       yy, 

'     •'•     from  (7i.)  ~f  +  TT?  = 
a2       d=62'  a2       ±.b2 


represents  P2Py 

Pi  is  called  the  pole,  and  P*P  its  polar,  with  respect  to  the  ellipse  or  hyperbola. 

322.  Examples.  —  (1.)  The  semi-axes  of  an  ellipse  are  4  and  2. 
Find  the  intercepts  of  the  polar  of  the  point  (  —  5,  6). 


Q,     0        - 

(2.)  Apply  the  data  of  the  last  example  to  an  hyperbola  by  using 
2-\/—l  instead  of  2  for  the  semi-conjugate  axis. 

Ans.xQ  =  -3~,  y0  =  —  f 

323.  Exercise.  —  When  the  pole  is  on  the  directrix,  the  polar  is  a 
focal  chord. 

Proposition  12. 

324.  Theorem.—  The  equation 


also  represents  the  locus  of  the  intersections  of  the  pairs  of 
tangents  to  the  conies 

J+JL^ 

a2      ±b*        ' 


124 


HYPERBOLA  AND  ELLIPSE. 


drawn   from    the    extremities   of   all   chords   which   pass 
through  any  fixed  point    (a*,  2/1). 


For,  let  Pj  be  the  fixed  point  through  which  all  the  chords 
pass,  and  let  Q^2  be  any  chord  .through  Pr  If  the  tangents  at 
Q1  and  Q2  meet  in  some  point  P2, 


then 


4. 


is  the  equation  of  the  chord  QtQ2  by  Art.  321,  and  at  P1  this 
equation  becomes 


Similarly,  if  the  tangents    at  the  extremity  of  another  chord 
through  P!  intersect  at  P3, 


But  (Art.  90) 


/v»/v» 


Eliminate  as  in  Art.  321,    .  *  .    -y-  +  ~^f  =^  represents  P2P3, 

which  is  the  locus  of  the  intersections  of  all  pairs  of  tangents  at 
the  extremities  of  chords  through  Pr 

PI  is  called  the  pole,  and  P2Pa  its  polar,  with  respect  to  the  ellipse  or  hyperbola. 


POLE  AND  POLAR. 


125 


325.  Examples. — Find  the   polars  of  the   following   points,  with 
reference  to  the  ellipse  x*  +  4yL  =  16,  and  the  hyperbola  x*  —  by*  =  16. 

(1.)         (2,  1.)  Ans.  y  =  -jx  +  4,  and  y  =  \x  -  4. 

(2.)         (#.-7.) 


(3.)        (2, 0.) 
(4.)        (0,  0.) 


28      7 
Ans.  x  =  8. 


. 

28      7 


326.  Exercise. — If  the  focus  be  the  pole,  the  directrix  will  be  the 
polar. 

Proposition  13. 

327.  Theorem.— The  equation 

x—x^      ±  62^! 
represents  a  normal  line  to  the  conic 

..2 

=  1 


a" 


at  the  point  (xly  y^). 


For,  since  (Art.  311)  i  = 

x     x 


is  one  form  of  the 


126  HYPERBOLA  AND  ELLIPSE. 

oy  -  /iy  C1   1J 

equation  of  the  tangent  line,     .'.     (Art.  128),   *     —  =  —  ~- 

X       X^  •    0  X-i 

is  the  equation  of  a  line  perpendicular  to  the  tangent  at  the 
point  (xlt  yj  —  that  is,  the  normal. 

328.  Cor.  —  If  y  —  0,  the  intercept  of  the  normal  is 


and  the  subnormal        =  x1  —  xl  (  1  —  -^=  —  )  =  —  —  x^  =  NM. 

\  a2  /         a2 

Similarly,  if  x==  0,  the  intercept  is  y  =  yx  (  1  --  )  =  CK. 


329.  Schol.  1—  In  the  hyperbola,  by  Arts.  283  and  328, 

FiN=  .Fi(7+  CN=  —  ae  +  #xlt 
and  (Art.  9)  F2N=  F2C  +  CN=  ae  +  e2xlf 

But  (Art.  304),  F^  =  -  a  +  exlt  and  F2Pi  =  a  +  e^. 

=  e.     . ' .     J*7!-^ :  J^A7^ :  :  -FiPi  :  -F2Pi. 


In  the  ellipse,  by  Arts.  283  and  328, 

F,N=  F,C+  CN=  ae 
JVF,  =  NC  +  CFS  =  ae 


.  •  .     By  geometry,  NP^  the  normal  of  an  ellipse,  bisects  F&F*  the 
internal  angle  of  the  focal  radii  of  PI. 


,.    Also,DP1N=NP1£. 

Subtract,  .  •  . 


.  •  .  the  tangent  bisects  FPiQ  the  external  angle  of  the  focal  radii. 
Similarly,  the  normal  of  the  hyperbola  bisects  the  external  and  the  tan- 
gent the  internal  angle  of  the  focal  radii. 


NORMAL.  127 


Conversely,  a  tangent  being  drawn  to  an  ellipse  or  hyperbola,  to  find 
the  point  of  contact.  Through  F*  draw  JF^Q  perpendicular  to  the  tan- 
gent, and  make  Q,S=SFv\  through  Q  draw  Qf\,  and  the  point  Plf  in 
which  it  cuts  the  curve  and  tangent,  will  be  the  point  of  contact. 

330.  Schol.  2.  f  —  Solve  the  equation  of  the  normal  for  y, 


Let 


y»  =       ± 


b-t 


77?- 


y(a2±b2m2) 
is  the  magic  equation  of  the  normal. 

331.  Exercises.  —  (1.)   Show  that  the   equation     2/Hl!  =  _^L- 

#  —  a?!      ±  6V, 

also  represents  a  line  through  the  pole  (xv,  T/J)  perpendicular  to  the  polar 
£^L  +  -^  =  7,   with  respect  to  the  curve  4  +  ~T,  =  1  (Art-  128)- 


(2.)  Show  that  in  the  ellipse,  P1N=  -  i/a*  —  e*x?, 


and  that 


.  •  .    (Art.  306)  ^2  =  P,N  .  P^. 


128 


HYPERBOLA  AND  ELLIPSE. 


(3.)  Show  that  in  the  hyperbola  also  ^/>2  =  P^N 


Proposition  14. 
332.  Theorem.—  The  equation 

x  cos  0     y  sin  0 

—+-T^= 

represents  that  diameter  of  the  curves 


a       =h 

which  bisects  the  system  of  parallel  chords  each  of  which 
makes  the  angle    0    with  the  axis  of  x. 


For,  from  Art.  100,    if  — : — rr  = T  =  I,     then  I  is  the 

sin  o       cos  u 

distance  from  any  point  (xlt  yj  upon  the  line     y  —  bl=x  tan  0, 
to  some  point  (x,  y)  at  the  intersection  of  this  line  with 


x2 

-*  + 


-*- 


.  • .     x  =  xl  + 1  cos  6,     and  y  =  y^  + 1  sin  6. 


Substitute  these  values  of  x  and  y,  in  (be.). 


(x,  +  I  cos  O) 


sn 


Expand, 


cos2  0      sin2  ff  \          x,  cos  g 


DIAMETERS.  129 


There  are  evidently  two  values  of  I.  Let  these  values  be  equal 
numerically,  but  of  opposite  signs  ;  then  the  point  (xlt  y^  must 
bisect  the  chord,  and,  by  the  "  General  Theory  of  Equations," 
the  coefficient  of  the  second  term  —  i.  e.,  the  coefficient  of  the  first 
power  of  I  —  must  vanish  ;  for  the  quadratic  is  the  product  of  the 
sum  and  difference  of  the  same  quantities. 

xl  cos  0       y.^  sin  6 
•'•'       ~~oT~       ~±¥~       ' 

in  which  (xv  y^  is  the  middle  point  only  of  any  chord.     Now, 
making  (xlt  yj  general, 

x  cos  6     y  sin  6 

«  _ 


is  the  locus  of  the  middle  points  of  all  these  parallel  chords  (Art. 
242),  and  is  evidently  a  right  line  through  the  origin  —  that  is, 
through  the  centre. 

'  333.  SchoL  —  When  xl  =  0,  and  yl  =  0,  we  have  the  distance  from 

1      /cos2  0      sin2  0yi 
the  origin  to  the  curve  -  —  I  —  -  —  |  --  —  1  »  ,     .  •  .     /is  always  real  m 

6  \      Cb  ~"~  U    I 

xi       IT  i  '         i  •    n     i  i       cos20      sin2  0 

the  ellipse,  and  is  real  in  the  hyperbola  when  -  --  ?.  e.,  when 

a2  01 

tan  0  <  HZ  -,  but  imaginary  when  tan  0  >  ±  -  —  that  is,  in  the  latter 
a  a 

case  the  line  in  the  given  direction  does  not  cut  the  curve 

4+-^  =  ^   butdoescut 
—   * 


. 

a?       —  b*  —a2       b* 

334.  Examples.  —  (1.)  The  semi-axes  of  an  ellipse  are  4  and  3  ;  find 
the  equation  of  the  bisectrix  of  a  system  of  chords  making  an  angle 
tan  ~'i/3  with  the  axis  of  x. 

Am-  =- 


(2.)  Find  the  same  for  the  hyperbola  whose  semi-axes  are  4  and 


130  HYPERBOLA  AND  ELLIPSE. 

Proposition  15. 
335.  Theorem.—  The  equation 

±b2 
m1m2  =  —  —  £--   expresses 

1st.  The  condition  that  the  diameter  y  =  m.2x  shall  bisect 
the  system  of  chords  y  —  bi  =  m^,  (in  luhich  6:  is  a  variable 
constant). 

2d.  The  condition  that  y  =  m&  and  x  =  mzx  shall  be  con- 
jugate diameters. 

3d.  The  fact  that  y  —  yi=m,(x  —  xl),  the  tangent  at  the 
extremity  (x^  y^)  of  the  diameter  y  =  m^,  is  parallel  to 
its  conjugate  diameter  y  =  m2x,  and  to  the  system  of  chords 

y  = 


4th.  The  fact  that  supplemental  chords  y  —  yi=ml(x  —  x^ 
and  y  +  7/1  =  w*  (x  +  »0  are  Always  parallel  to  conjugate 
diameters  y—^i^x  and  y  =  m2x, 


1st.  The  equation  of  the  diameter  (Art.  332), 
x  cos  6     y  sin  6 

--  L*  -  —0 

a2  ±b2         ' 

±b2 
solved  for  y,  is        y  —  ~  —  T"  x  cot  #>  or  y  =  mp.    The  equation 

CJL 

of  the  chords  it  bisects  is  y  =  x  tan  0  +  bl}  or  y  =  m^x  -f  blt 

±b2 


2d.  "When    y  =  x  tan  6  -\-  bl}    or    y  =  m-p  +  bv    is  the  system 

±b2 
of  chords,  then  (by  1st),  y  =  —x.  -7-  -  7  ,      or  y  =  m2» 

is  their  diameter  —  i.  e.,  the  coefficient  of  x  for  the  diameter  is 
obtained  from  the  coefficient  of  x  for  the  chord  by  multiplying  its 

reciprocal  by  —  —  y  . 


CONJUGATE  DIAMETERS,  ETC. 


131 


±b2 

Similarly,  when       y  =  —  x'  -JT — ^  +  b.2,  or  y  —  m^c  +  62, 

ci"  tan  0 

is  the  system  of  chords ;  multiply  the  reciprocal  of  the  coefficient 
of  x  by  -  =Y~,  .  * .  y  =  x  tan  0,  or  y  =  m^  is  the  diameter, 
which  is  parallel  (Art.  128)  to  the  chords  y  =  m^x  +  blt 

.  * .     (Art.  242)  y  =  mlx  and  y  =  m^c  are  conjugate, 

±62 

3d.  Now,  y  —  y\  =  —  ~~i — l  (x  —  x^,  or  y  —  yx  =  m2  (x  —  arj  rep- 
resents (Art.  317)  the  tangent  at  (xlt  yj,  and  y  =  —x,  or  y  =  mlx, 

1 

is  the  diameter  through  (#1;  yj, 


4th.  When  P  is  any  point,  y  —  yl  =  ml  (x  —  x^  represents  PPr 
Also  (since  y2  =  —  y^,   y  +  yx=  m2  (a?  +  a^)  represents  PP2. 
Multiply,         . ' .     y2  —  yx2  =  m^j  (or2  —  a^j2)  .  .  .  (m.) 

x2       y2 
Again,  if  Pl  and  P2  are  on  the  curve,  then  -y  +  ~~f*  —  lt 


x2        y2 

—  +  — — =  / 

a2+±62     2' 


and  if  P  or  (xt  y)  is  also  on  the  curve, 

Subtract,   .'.    - — ~--f- — ^-=^  or  y2-y12  =  -=1^- (a;2-^2), 
a2  ±  b2  a2 

which  (Art.  112)  is  the  equation  of  the  two  chords  PPl  and  PP2 
respectively.    Divide  the  equation  by  equation  (m.)9 

±b2 


132  HYPERBOLA  AND  ELLIPSE. 

336.  Schol.  —  The  conjugate  diameters  in  the  ellipse  both  cut  the 
curve  (Art.  333),  and  since  m^m^  =  --  -,  if  m±  is  the  tangent  of  an 

acute  angle  it  is  +  ,  and  m2  is  —  ,  and  is  then  the  tangent  of  an  obtuse 
angle,  and  vice  versa.  In  the  circle  a  =  b,  .  •  .  m^n*  =  —  1 
(cf.  Art.  123). 

The  angle  between  the  axis  of  x  and  the  conjugate  diameters  in  the 
hyperbola  are  both  acute,  or  both  obtuse,  since  nfi^m^  =  -\  —  -,  and  if 


Wi  >  ±  —  ,    •  *  -    wi2  <  ±  -.     Hence  (Art.  333),  only  one  of  the  con- 
a  a 

jugates  intersects  the  hyperbola. 

x*       yz 

337.  Examples.  —  (I.)  Given  an  ellipse  --  \-£-  =  lt  and  a  diame- 

J.O          t/ 

ter  making  an  angle  of  30°  with  the  axis  of  x  ;  to  find  its  conjugate. 

Ans.  16y  =  — 
(2.)  Find  the  equation  of  a  tangent  to  the  same  ellipse  parallel  to 

the  line  -£--1  =  1.  Ans.  y  =  2x±  8.644. 

o.o      7 

X*          7/2 

(3.)  Perform  the  same  operations  with  the  hyperbola  —  —  *—  =  1. 

It)          a 

338.  Exercise.  —  Construct    conjugate   diameters;    also   a    tangent 
parallel  to  a  given  line. 


Proposition  16. 
339.  Theorem.— The  equation 

represents  an  ellipse  or  hyperbola  referred  to  conjugate 
diameters  as  co-ordinate  axes;  in  which  a?  and  ±:b* 
are  the  squares  of  the  semi- conjugate  diameters. 

For,  change  the  direction  of  the  axes  in  the  curve 

^+±72=^' 

By  Art.  71,  x  —  xf  cos  6  +  yr  cos  01;   and    y  =  x'  sin  6  +  ?/'  sin  Or 


CONJUGATE  DIAMETERS  AS  AXES. 


133 


(xr  cos  6  +  y'  cos  O^2     (xr  sin  0  +  yr  sin  0^ 

*>  .TO 


or 


Y'        Y 


But  since  these  are  conjugate  diameters,  by  Art.  335, 

±  b2                               cos  6  cos  6l     sin  0  sin  6l 
tan(9  tan^  +  — 7-  =  ^,    .'.   by  trig.-     -^ + — ^ 


(2 
y 


±0 

cos2  0     sin2  0      7  cos2  0,       sin2  ^         1 

=  — ;,    and  '    ' 


But(Art.333),^^-  +  ^-  =  -2, 


x'2  ,     y'2 


340.  Exercise. — Prove  that  -^-+  ^    =-?  is  the  equation  of  the 

«i2       ±b? 

tangent  referred  to  conjugate  diameters. 


Proposition  17. 
341.  Theorem— The  equations 

Til  Til 

•^l  ,       </2  J  2  _i_    "l 

—  =  ±-r,    and    —  —  ± -r 
a  b  a  b 

express  the  relations  between  the  co-ordinates  of  the  ex- 
tremities   (xi,  2/1)    and    (x2,  y.2}    of  conjugate  diameters. 


134 


HYPERBOLA  AND   ELLIPSE. 


For,  if  —  =  —  is  the  equation  of  a  diameter  whose  extremity 

1          J  1 

is  (#1,  yj,  by  Art.  335,  2d,  its  conjugate  is  —  \-  +  ~L~~h  —  0. 


.  •.    at  fe  y2),  we  have 


=  0,  and 


Eliminate  y2,  .  •  .       +  7  =  -?,  or        •        -       +       =  -?  J 

a2      ±6V2  a2        2±62      a2 


3;.  •.  — 7 

•         •  o  o  -i  • 

a2        2       ' 


Similarly,  it  may  be  shown  that  —  =  ±  ~. 


Proposition  18. 
Theorem.— The  equation 


is  an  equation  of  condition  of  conjugate  diameters. 

For,         a2  =  x,2  +  y?, 
and  b2  =  x2  +  y2 ;  and  taking 
the  values  of  x2  and  y2  from 

b2  a2 

Art.  341,    6V  =  -7  x,2  +  -TJ  yA 


2          =   2 


Proposition  J.9. 
343.  Theorem.—  The  equation 


expresses  the  square  of  the  length  of  a  semi-diameter  con- 
jugate to  Oi,  when  /?x  a-w/c?  |02  are  ^^e  focal  radii  of  the 
extremity  of  ai. 


ECCENTRIC  ANGLE  OF  THE  ELLIPSE. 


135 


For,  ±  bf  =  a2  ±  b2  -  of,  by  Art.  342, 


=  a2  ±  b2  -  (x?  +  y?}  =  a2  ±  b2  -  x,2  -  — ^  (a2  -  av2), 


But  (Arts.  304,  306) 


=  ±(a2-  e*x2).    .  •  .    ^2  =  ±  b2. 


Proposition  £0.f 
344.  The  equations 

x  y 

-  =  cos  (p   and    7  =  sin  c> 

a  b 

together  represent  an  ellipse;  in  which    <p    is  the  eccentric 
angle. 

IT  77 

For,  squaring  and  adding  the  equations,  we  have   —  +  77  = 
sin2  <p  +  cos2  <p  =  l,  by  trig.     This  equation  represents  an  ellipse, 

/y»  ft 

hence    -  =  cos  <p,  andy  7  —  sin  <py  must  together  represent  the  same. 


345.  Schol.  1. — The  eccentric 
angle  of  PI  is  JT(7P2 ;  for  since 
the  ellipse  is  a  projection  of  the 

qj  /y 

circle,  by  Art.  297,  —  =  — ,  and 
by  sim.  tri's     ^  =  — ±  =  -?—. 


similarly,  if  (7P2  =  a,  then  -  =  cos  <f>. 


136 


HYPERBOLA  AND  ELLIPSE. 


346.  Schol.  2.  —  Since    in    the    parallelogram 


we    have 


This  principle  is  employed  in  the  construction  of  the  trammel  or 
elliptic  compasses.  For,  if  MI  and  N  run  upon  the  axes  as  guides,  any 
point  P!  of  the  line  MiN  will  describe  an  ellipse.  The  point  halfway 
between  M  and  N  describes  a  circle,  a  particular  case  of  the  ellipse. 

347.  Schol.  3.— The  point  P! 
corresponding  to  the    eccentric 
angle  <p  is  most  easily  constructed 
as  in  the  figure.     This  affords  a 
good    method    of   constructing 
an  ellipse  by  points;  for  since 
yc  =  a  sin  <p,  and  ye  =  b  sin  ^>, 
we  have  by  subtraction 

Hence,  draw  two  concentric  circles  whose  radii  are  respectively  a 
and  b,  and  from  the  points  Qv  and  N  where  any  radius  cuts  the  two 
circles,  draw  parallels  to  the  axes  of  y  and  x  respectively  as  repre- 
sented ;  their  intersection  will  be  a  point  of  an  ellipse  whose  semi-axes 
are  a  and  b. 

/v»  fit 

348.  Exercise. — Show  that   -  cos  <pl  +  —  sin  <f>i=l  is  the  equation 

a  b 

of  a  line  tangent  to  the  ellipse  at  a  point  whose  eccentric  angle  is  ?lt 


Proposition  21.^ 
349.  T7ieorem.—The  equations 

x  y 

-  =  sec  a>    and    7  =  tan  <p 

a  o 

together  represent  an  hyperbola;  in  which    <p    is  the  eccen- 
tric angle. 

y?      y2 
For,  by  squaring  and  subtracting  —  —  -77  =  sec2  <p  —  tan2  <p  =  1, 

CL  O 

by  trig.,  which  represents  an  hyperbola ; 


•Ju  U 

hence        -  =  sec  <p,  and  '7  =  tan  <f>,  must  represent  the  same. 


ECCENTRIC  ANGLE  OF  THE  HYPERBOLA. 


137 


350.  Schol.  1.—  The    eccentric    angle    of    P  is    XCPZ.      For,   if 

#2       7/2 
<7P2  =  «,  then     <?J/=  x  =  a    sec    ^  ;     and      --  -77  =  -Z      becomes 


sec 


_ 
—^—  =  lt  or      =  |/'  sec2  <p  —  1  =  tan  <f>,  by  trig. 


Also,  if  CP3  =  £, 


-=tan 


then 


.  •  .    also  P3M3  = 

351.  Schol.  2.—  The 

eccentric  angle  affords  a 
method  for  constructing 
the  hyperbola  by  points. 

Draw  two  concentric  circles  with  the  radii  a  and  5,  and  from  the 
points  where  any  radius  cuts  the  two  circles  draw  the  tangents  P2M, 
P3M3.  Then  M^N=M,Pz=y  and  CM=x.  Through  M  and  N 
draw  parallels  to  the  axes  y  and  x  respectively  ;  their  intersection  at 
P  is  a  point  of  the  hyperbola. 

352.  Exercise.  —  Show  that  the  line 


x 

-  sec 
a 


, 

—  £  tan  <?i  — 
b 


is  a  tangent  to  the  hyperbola  at  the  point  whose  eccentric  angle  is  ^. 

353.  Schol.  —  The   eccentric   angle   of  any   point   P  of  the   conic 

#2         y1 

—  =  —  -  —  =  1  is  included  between  the  axis  of  x  and  that  radius  of 

a2      ±  b'2 

the  circle  xz  +  y2  =  «2  which  has  the  same  subtangent  as  P  (Art.  313). 


Proposition  22.-\ 
354.  Theorem.—  The  equation 


is  also  the  equation  of  condition  for  conjugate  diameters  of 
the  ellipse  and  hyperbola;  in  which  <pl  and  y^  are  the 
eccentric  angles  of  the  extremities  of  the  conjugates. 


138  HYPERBOLA  AND  ELLIPSE. 

/v»  n  i 

1st.  In  the  ellipse  (Art.  344),     -  =  cos  <p  and  7  =  sin  <p, 

and  (Art.  335),  m.m^     ^. 

y^      b  sin  <pl      b 

But  by  trig.,       ml  —  —  = =  -  tan  tplt 

xl      a  cos  <pl     a 

?/2      b  sin  <p2      b 

and  m2  =  —  = =  -  tan  <p2, 

x2      a  cos  <p2      a 

b2      b2 

*  /Wl     /W)       — —   - *f  Q  Tl       //>        4"  Q  T"»       /fl 

•         •  A/t/i  //t/9  9    9      Ltlll     U/|       Lclll     1^9* 

a^      aj 

i 

.  • .     tan  ^  tan  <f>2  =  —  l.     .  • .     (Art.  123),        ^ x  —  y?2  =  ^- 

/^Y     /^\2          T  y 

2d.  In  the  hyperbola  (-)  —(7)  =-?,  L=sec  «p,  and  7  =  tan  CP. 
^r  \a/      \b/  a  b 

Similarly  in  the  conjugate  hyperbola  TTJ  —  (-J  =1, 

r  =  cosec  <p,     and     -  =  cot  <p. 
yl      b  tan  ^      5    . 


Also 


cosec 


and  m2  =  —  =  —  -  sec  <f>2. 

2      x2        a  cot  tf  2       a 

For,  if  y  =  ra1a;  cut  the  primary  hyperbola,  then  y=m<p,  cuts  the 

2 
conjugate,  by  Arts.  333  and  337,  when 


b2      b2 
,m2  =  —2  =^  sin  ^  sec  (f>2; 

sin 

=/j     .-.    by  trig. 


CONJUGATE  DIAMETERS. 


139 


355.  Schol. — This  enables  us  (see  fig.  of  Art.  347)  to  construct  con- 
jugate diameters  in  an  ellipse  by  drawing  two  radii  of  the  circle  r  =  a 
at  right  angles.  The  ordinates  of  their  extremities  Qi  and  Q.2  will  give 
the  extremities  PI  and  P.t  of  the  conjugate  diameters. 

Conjugate  diameters  in  the  hyperbola  may  be  constructed  as  follows: 


Draw    two    circles    with    a    and    b    for    radii,    and    let    the    angle 

nd  P.2,  having  the  same 
and  B^,  will  be  the 


XOY=  XCQl  +  XCQ*  =  -.     The  points  P^  and  P.2,  having  the  same 


subtangents  M^M^  and  JV^  as  the  arcs 
extremities  of  the  conjugate  diameters. 


Proposition  23. 
356.  Theorem.— The  equation 

albl  sin  p  =  ab 

expresses  the  fact  that  the  tangents  through  the  extremities 
of  any  pair  of  conjugate  diameters  enclose  the  same  area ; 
in  which  ft  =  ol  —  0^  is  the  angle  between  the  semi-conju- 
gate axes  di  and  bi. 

Let  — j-  +  ~h  =  1  be  the  tangent  line  through  (xl}  y^,  the 


extremity  of  the  diameter  y  = 


i 
a2 


;,  and  parallel  to  the  diameter 
7  =  0. 


140  HYPERBOLA  AND  ELLIPSE. 

By  Art.  142,  the  length  of  the  perpendicular  from  the  origin  on 

the  tangent  is  p= 


But  (Arts.  341,  342), 


Now,  in  the  figs,  of  Arts.  347  and  355  the  area  of  the  parallelo- 

gram CP^DPz  =pblt  but  phi  =  ab. 

Again,  by  trig.,  CP1DP2  =  a1^1  sin  /9     .  •  .     ab  =  a^  sin  /?. 

Now  4<zb  =  rectangle  of  the  axes. 

And  4a}k>i  sin  ft  =  rectangle  DD'  . 


357.  Schol.-Ky  Art.  356,  sin'  0  -  ~  = 


-,^  A  2,2  . 

+  &i2)2  -  («i2  -  V)2  ' 

.  •  .     (Art.  342)  sin2  /?  =  I,  _       2  for  the  ellipse.  .  .  (n.) 


the 


In  the  ellipse,  when  a:  =  Jj,   sin  ^  —  ~  ~       is  a  minimum  value  of  («.). 

These  are  the  equi-conjugate  diameters  ;  and  since  the  ellipse  is 
symmetrical,  Wj  =  —  m2  in  the  equation  m1m2--=  --  ^, 

CL 

and     .*.     w=±—  for  the  equi-conjugate  diameters.     .*.     they  lie 
a 

in  the  diagonals  of  the  rectangle  of  the  axes. 

In  the  hyperbola,  sin  0  =  0  is  a  minimum  value,  for  a^  increases  with 

bi,  and  the  denominator  of  (o.)  can  be  made  infinite.    These  conjugates 

b'z 

are  infinite,  and  coincide.     Since  m^m^  =  —  , 

a2 

when  ml  =  m.2,  then  m=±-.  These  are  self-conjugate  diameters, 
and  will  be  shown  to  be  asymptotes  —  i.  e.,  tangents  at  an  infinite  dis- 
tance from  the  origin. 

Also  (Art.  284),  e  =  sec  AlCDl  (fig.  of  Art.  358). 


ASYMPTOTES  OF  THE  HYPERBOLA. 


141 


Proposition  24. 
358.  TJieorem.—TJie  equation 


represents  the  asymptotes  of  an  hyperbola  and  its  conjugate 
hyperbola. 

For,  the  equation 
of  the  tangent  line 

.g  x^+yy^=L 


which  is  the  inter- 
cept on  x.     If  x  =  0, 

yn  —  —     -,  which  is 

2/i 
the  intercept  on  y.     If  the  co-ordinates  of  the  point  of  tangency 

are  0^  =  00  and  y^  =  oo;  then  XQ  =  0  and  yQ  =  0.    .  • .    the  tangent 

at  a  point  infinitely  distant  passes  through  the  origin. 

Also  (Art.  342),  of  =  x?  +  y?,     .' .     al  =  oo)  and  then  (Art.  357) 

b  b 

m  =  ±:-,     .'.     y  =  mx,  or  y  =  ±-/is  at  once  a  diameter  and 

tangent  at  infinity.     The  same  may  be  proved  for  the  conjugate 
hyperbola.     The  equation  y  =  ±-x  may  also  be  written 

(x     y\  fx     y\  xz      v2 

-I     ~  +  I  )=°,     or     —-ij^0- 
a     b/  \a     b]  a2       b2 

359.  Schol. — The  asymptotes  are  the  diagonals  of  the  rectangle  formed 
by  the  tangents  at  the  vertices — i.  e.,  the  rectangle  of  the  axes. 

360.  Exercise. — Prove   that  y  =  ±  —x  is  the   equation   of    the 

al 

asymptotes  referred  to  conjugate  diameters  as  axes. 


142 


HYPERBOLA  AND  ELLIPSE. 


Proposition  25. 

361.  Theorem.—  The  equations 


xy  =        -j- 

represent  an  hyperbola  and  its  conjugate  referred  to  the 
asymptotes  as  axes. 


x' 

.  •.   cos      = 

X 

and 


. 
sin  x  --sin  £; 

.  •  .     (-Art.  71) 

x  —  (x'  +  yr)  cos  d 

and 

y  =  (y'-xf]  sin  6. 

But  (Art.  357)  tan  0  =  -,     .  '  .     by  trig.,  cos  0  = 


,2. 

~T  0    ) 


and 


= 


b  a(x'+y>} 

hence  ^=2       8  d2/- 


a:2      y2 
Substituting  these  values  in    —  —  jr  = 


we  have 


(x'  +  y'Y     (y'-xj 


1, 


whence 


a? 


Similarly,  substitute  in  —  2  +  TJ  =.?  for  the  conjugate  hyperbola, 


4    ' 


ASYMPTOTES  AS  AXES. 


143 


X  I/ 

362.  Schol.  1. — The  equation h  —  =  2  represents  the  tangent 

at  (a?lf  yj)  referred  to  the  asymptotes.    For,  the  line  through  (a:,,  yj  and 
(a?t>  y2)  is  (Art.  90), 


—  y  \_yi-  y\ 


But 


yl  =  a:*  y2  = 


y— yi 


a;— 


=.2L.     Place  *,  =  :*,  then  -£  +  -£  =  *. 


363.  Schol.  2. — In  this,  if  y  =  0,  xQ  =  2xl  =  intercept  on  x ;  and  if 
x  =  0,  yQ  =  £yi  =  intercept  on  y.  .  • .  a:0y0  =  4x\y\  =  a2  +  62 ;  . ' .  the 
product  of  the  intercepts  is  constant. 


364.  Schol.  3.—  By  trig.,  area  CPl 
=  x^i  sin  £0  =  -  sin  20,  since 


—      - 
4 


=      2       sin  0  cos  e  =~  (Art.  361). 
.  * .     area   GA\  =  area   CPi  =  — . 

365.  Schol.  4. — Since  x^  =  —  and 
yj  =  — ,  the  point  of  contact  (by  sim.     /C       MIK 


tri's)  bisects  JOT,  the  portion  of  the  tangent  between  the  asymptotes. 

Since  OPl  bisects  MN,  it  bisects  any  parallel  as  HK  at  Q. 
.  • .  HQ  =  QK.  But  since  CP}  is  a  diameter,  PQ  =  QAt.  . ' .  HP 
=  A^.  Hence  to  construct  an  hyperbola  by  points,  draw  a  series  of 
lines  through  A±,  and  in  each  line  make  HP=AiK,  then  the  locus 
of  P  is  the  hyperbola. 

366.  Examples. — Construct  the  loci  represented  by  the  following 
equations. 


144 


HYPERBOLA  AND   ELLIPSE. 


(1.)  —  +  +-  =  1,  in  which  0  =  60°. 

JL  ts 

(2.)  4x*  —  y*  —  4,  in  which  0  —  30°. 

Find  the  equation  of  a  tangent  to  each  of  the  above  loci  at  a  point 
whose  abscissa  is  2. 


Ans.     (1.) 
(20 


4x 


(3.) 

(40 
(5.) 


17x 


=  68. 


367.  Exercise. — Prove  that  the  asymptotes  are  the  diagonals  of  the 
parallelogram  formed  by  drawing  tangents  through  the  vertices  of  any 
pair  of  conjugate  diameters. 


Proposition  26. 
368.  TJieorem.—The  equation 

I 
'      1  +  e  cos  6' 

represents  an  ellipse  or  hyperbola ;  in  which   p    and    6    are 
the  polar  co-ordinates,    0    being  measured  from  the  nearest 

vertex,  and-  I  —  — —  is  the  semi-latus  rectum,  the  pole  being 
a 

cut  one  of  the  foci. 

For,  in  the  ellipse  (Art.  307), 

Pi  +  p2  =  %a> and  by  t-rig-» 


PI  = 

Eliminate 


=  —  1//?22 

Squaring  and  reducing, 


cos  u. 
p2       **a 
,  cos  6. 


±b2 


e  cos  6        a       1  +  e  cos 


(Art.  288). 


POLAR  EQUATION.  145 


Let  0  =  -,  then,  f>3  =       =1,     .'.     p  = 


a  1  +  ecosd' 

The  same  may  be  proved  for  the  hyperbola. 

369.  Schol. — The  equation  p  =  -j —        —  represents   an   ellipse, 

and  the  equation  p  =  =—± represents  an  hyperbola,  in  which  0  is 

1  —  e  cos  0 

measured  as  usual  from  +  x,  and  the  sign  -f  or  —  is  used,  according  as 
the  right  or  left  hand  focus  is  the  pole. 

370.  Examples. — Construct  the  curves  represented  by  the  follow- 
ing equations. 


w 

(2.) 
r.q  ^ 

3  +  i/5  cos  0 
1 

2  +  tan  -  cos  0. 
S 

4 

3  +  cos  0  +  4  cos  0  sin—  • 

Construct  the  polar  of  the  point  (4,  6)  with  reference  to  the  follow- 
ing loci. 

(4.)  x*-y*  = 


(5.)  x*  +  y2  =  4,  in  which  a>  =  V  =  60°. 

Ans. 
(6.)  Interpret  the  equation  x*  —  y2  =  x  +  y. 

371.  Exercises.  —  (1.)  Prove  as  in  the  parabola  that  the  focal  polar 

equation  of  the  tangent  line  is  />  =  -  ;  -  -,  in  which  0  is 

e  cos  0  +  cos  (0  —  9$ 

measured  from  the  nearest  vertex. 

(2.)  Prove  by  transformation  that  the  central  polar  equation  of  the 

ellipse  and  hyperbola  is  p  =  -  ^  -  -  —  . 
10 


CHAPTER    VIII. 
GENERAL   EQUATION   OF    THE   SECOND   DEGREE. 

Proposition  1. 

372.  T7ieorem.—The  most  general  equation  of  the  second 
degree,  viz. : 

Ay?  +  2Hxy  +  By2  +  %Gx  +  2Fy  +  C=  0*  .  .  .  (a.) 

always  represents  one  of  the  conic  sections  when  A,  B,  C, 
F,  G  and  H  are  any  real  constants. 

The  truth  of  this  proposition  will  appear  as  the  result  of  the 
succeeding  propositions,  in  the  following  manner.  We  shall 
move  the  origin  either  to  the  centre  or  to  the  vertex,  and  then 
change  the  direction  of  the  axes  of  x  and  y  so  that  the  axis  of 
x  shall  coincide  with  the  principal  axis  of  figure  of  the  curve ;  it 
will  then  appear  that  («.)  is  reduced  to  a  form  identical  with  one 
of  those  before  found  to  belong  to  the  conic  sections. 

373.  Schol.  1. — The  same  result  would  follow  whether  (a.)  be  in 
oblique  or  rectangular  co-ordinates.     For,  if  it  be  in  oblique  co-ordi- 
nates, and  be  transformed  to  rectangulars,  the  co-efficients  A,  B,  C,  etc., 
will  be  changed,  but  the  general  form  will  remain  the  same  (Art.  229). 
We  shall,  therefore,  without  the  loss  of  generality,  consider  the  case  of 
rectangular  axes. 

374.  Schol.  2. — In  this  discussion  A  is  supposed  to  be  a  positive 
quantity. 


*  We  here  adopt  the  coefficients  in  ordinary  use  at  present.  x 

It  may  be  useful  to  notice  their  symmetry  as  expressed  in  y 

the  square ;  e.  g.t  B  is  the  coeff.  of  y2,  2F  of  y,  2G  of  x,  etc.  1 
146 


CO-ORDINATES  OF  THE  CENTRE.  147 

^Proposition  2. 
375.  Theorem.—  TJie  equations 

BG-HF  AF-HG 


express  the  co-ordinates  of  the  centre    (x0,  y0)    of  the  locus 
represented  by  equation  («•). 

For,  let  us  move  the  origin  to  (x0,  y0)>  whose  co-ordinates  are 
yet  to  be  determined.     From  Art.  23, 

x  ——  x    i™  XQ}     and    y  —  i 
.  • .     (a.)  becomes 


-  G)xf  - 

Now,  if  (XQ,  T/O)  is  the  centre,  (6.)  must  be  of  such  -form  as  to 
remain  unchanged,  whether  we  substitute  for  x'  and  y',  +  x'  and 
+  y',  the  co-ordinates  of  one  extremity  of  a  diameter,  or  —xrt 
and  —  y',  the  co-ordinates  of  the  other  extremity  of  the  same 
diameter,  since  then,  the  origin  evidently  bisects  the  distance 
between  (x' ',  y'}  and  (—x/  —y').  That  such  may  be  the  case, 
there  must  be  no  terms  of  the  first  power  in  (6.) — i.  e., 

from  which  by  elimination  we  find 

_BG-HF  ^AF-HG- 

X°~  H2-AB'-(Ct}>  and2A>-  H2-AB  '" 


376.  Schol.  l.—li  H2  —  A£  =  0,  the  centre   of  the  curve   is   at 
infinity  —  i.  e.,  it  has  no  centre  ;  but  if  JET2  —  AB>0,  it  has  a  centre. 

377.  Schol.  2.  —  It  may  he  noticed  that  the  transformation  result- 
ing in  (6.)  does  not  change  A,  B  or  H,  and  that  the  new  constant  term 

Axfa-QEx&t+Byfa  Gxo+2Fy*+Q-=  Cr  .  .  .  (e.) 
is  of  the  same  form  in   XQ,  y0   as  (a.)  is  in  x,  y. 


148  GENERAL  EQUATION  OF  SECOND  DEGREE. 

378.  ScJiol.  3.  —  Hence    the    result   of  the   transformation   to   the 
centre  may  be  written 


379.  Examples.  —  Find  the  co-ordinates  of  the  centres  of  the  curves 
given  in  Art.  391,  and  the  values  of  G'. 


Proposition  3. 
380.  TJieorem.—The  equation 


A-B 

expresses  the  value  of  the  angle  0  through  which  the  co- 
ordinate axes  x  and  y  in  equation  (a.)  must  be  turned  to 
cause  them,  to  be  parallel  to  the  axes  of  the  curve. 

To  turn  the  axis  through  any  angle,  6,  we  place  (Art.  80) 
x=x"  cos  6  —y"  sin  6,   and    y  =  x"  sin  6  +y'f  cos  0. 
Substitute  in  (a.)     . ' . 

(A  cos2  6  +  £J7sin  6  cos  6  +  B  sin2  0)  x"2 
+  2[(B  -  A)  sin  6  cos  6  +  ^T(cos2  6  -  sin2  6}]  x"  y" 
+  (A  sin2  0  —  &5"sin  6  cos  6  +  B  cos2  6)  y"2 

>-£sin 


This  may  be  written 

A V'2  +  2HWy"  +  B'ym+  8G'x"  +  2Ffy"+C=  0 . . 
Now,  if  H'  =  0,  or  H(cos?0-sm2d)-  (A  -B}  sintfcostf  =0, 
H         sin  6  cos# 


then, 


-B    cos20-sin20' 


AXES  OF  FIGURE.  149 


By  trig.,     sin  20=2  sin  6  cos  0,   and   cos  26  =  cos2  0  —  sin2  0. 

H         j      sin£0      i 
Substitute,        .  • .     T7~» =  *  '  "  "^  =  *  tan  ^  •  •  <  •  (*•)• 

J\.         x3  COS  /v " 


Hence  (ft.)  becomes 

^V'2  +  £y'2  +  ££V'+£^y'  +  <7=0.  .  .  .  (I.). 
Now  complete  the  square  with  respect  to  both  x"  and  y". 


or 


'\2 
\  _i_  7?/  ?/r  j  __    —  —  _i  --  —  C=  —  (y 

'}  H       \y     ^  5V          '          '  ; 

G^'V     /         .F\a 

^-)  r+^ 


which  (Art.  296)  represents  an  ellipse  or  hyperbola  referred  to 
axes  parallel  to  the  axes  of  the  curve,  when  neither  A'  nor  Br  is 
equal  to  zero. 

381.  Schol.  1.  —  By  a  similar  transformation  of  (/.)  its  axes  of  x  and 
y  will  become  coincident  with  the  axes  of  the  curve  ;•  call  the  result  (</.)'• 
By  trig.,  sin2  6=^(1  —  cos  20),  and  cos2  0  =  ^(1  +  cos  20). 

Substitute  these  relations  and  those  above  in  (fir.)', 

.  •  .    \(A  +  B  +  2H  sin  00  +  (A  -  B)  cos  20)a!* 
-  4)  sin  20  +  2H  cos  2d}x"f 

d-(A-  B)  cos  2%"2  +  C'  =  0. 


If  tan  20  =  ~h     then,  as  in  Art.  380,  A'af*  +  By"*  +C'  =  0, 


which  is  the  central  equation  of  some  conic,  if  neither  A'  nor  B  is  equal 
to  zero. 

382.  Examples*  —  Find  the  values  of  0  for  the  .  curves  given  in 
Art.  391. 


150  GENERAL  EQUATION  OF  SECOND  DEGREE. 

Proposition  4. 
383.  Theorem.— The  equations 

and    A'  +  B'  =  A  + 


express  relations  which  are  invariable  in  any  equation  of 
the  second  degree,  whatever  may  be  the  position  of  the  axes. 

For,  evidently  the  transformation  in  Art.  375  does  not  change 
A,  B  or  H,  but  that  of  Art,  380  gives  us 

A'  =  \  [(A  +  B)+(A-B)  cos  20  +  %H  sin  20], 

2Hr  =  (B—A)  sin  20  +  2H cos  20, 
B'  =  $[(A  +  B)  —  (A—B)  cos  20 - 2H sin  20], 

.-.    A'+B'=A+B.  .  .  .  (p.). 
Also,  H'2-A'B'  = 

{[(A  -  B)  sin  20  -  2H  cos  20]*  -?(A  +  B)2 

+  {[(A- B)  cos  20  +  2H  sin  26]*. 
By  trig.,  sin2  20  +  cos2  20  =  1, 

.  • .    4(Hf2  -  A'B')  =  (A-  B)2  +  4H2  -(A  +  B)\ 
.-.    H'2-AfB'=H2-AB.  .  .  .  (q.). 

384.  Schol.  1.— Since,  when  (Art.  380)  tan  20  =  — ,  or  H'  =  0 

^cL  —  .0 

the  axes  of  x  and  y  are  parallel  to  those  of  the  curve,  (q.)  then  becomes 

9  TT 

385.  Schol.  2.— If        tan  20  =  . 


1  1        •  *  /T\f\  Lclll     X/C/  -, 

by  trig.,      sin  *0  =  — —   ——  and  cos  **  = 


tan2  Mf  i/(2  +  tan2 

Substitute,   .  • .   sin  ^  = 

and  cos  20  = 


THE  CRITERION  H*-AB.  151 

Substituting  these  values  in  the  values  of  A'  and  1?, 
•          ' 


and  B  =  i[(A  +  B)  -  i/(A  -  Bf  +  (£57]. 

386.  Exercise.—  Prove  that  in  Art.  380 

ABC-VFGH+  AF*  +  BGP  +  Off* 


C'  = 


H*-AB 


^Proposition  5. 

387.  Theorem.— The  invariant  expression 

H2-AB 

is  also  the  criterion  for  determining  which  particular  curve 
is  represented  by  a  given  equation  of  the  second  degree. 

The  curve  is  an  Ellipse  if  H2  —  AS  <  0. 
The  curve  is  a  Parabola  if  H2  —  AB  =  0. 
The  curve  is  an  Hyperbola  if  H2  —  AB  >  0. 

For,  when  tan  £<?=",  then  (Art.  384), -A'B'=H2-AB. 

If  A'  and  Bf  have  like  signs,  then  equation  (m.)  or  (n.)  evi- 
dently represents  (Art.  288)  an  ellipse, 
and  then,     -  A'B'  <  0,  .  • .     from  (r.)  H2-AB<  0. 

If  A'  and  B'  have  unlike  signs,  then  eq.  (w.)  or  (n.)  represents 
(Art.  281)  an  hyperbola,  and  —A'B'  >  0,  .'.  (r.}  H2-AB>  0. 

If,  however,  -A'B'=H2-AB  =  0,  then  A'  =  0,  or  B'=0. 
Suppose  A'  —0;  by  Art.  376,  the  centre  is  in  this  case  at  infinity, 
and  equation  (I.)  becomes 

B'y' f2  +  QF'y"  +  2G-fx"  +  C=  0, 
I         jF'\2        BG-'I  C         F'2 

which    (Art.  243)   represents   the   parabola  y2  =     -  -^7  x,  whose 
origin  has  been  moved  by  using  (Art.  23)  equations, 


152  GENERAL  EQUATION  OF  SECOND  DEGREE. 


B'C-F'Z  Ff 

when  x°  =  ~7r~  2/o~7- 


388.  Schol.  l.—li  —A'B  =  H*-AB<0. 

C'  C' 

When  --  -r,>0,  and  —  -  >  0,  (n.)  represents  a,  real  ellipse  (Art.  288). 

A  JD 

When  C'  =  0,  the  ellipse  reduces  to  two  imaginary  right  lines. 

C'  C' 

When   --  —  <  0,    and    —  —  <  0,  the  ellipse  is  imaginary  (Art.  301). 
A.  B 

If  A'  =  B',  the  ellipse  becomes  a  circle. 

389.  Schol.  £.—  If  —A'B  =  H'i-AB  =  0, 

and  A'  =  0  only,  (s.}  represents  a  real  parabola.  If  also  Gr  =  0,  then, 
when  Fn  —  B'C^>  0,  the  parabola  becomes  two  real  parallel  right  lines  ; 
when  F'*  —  B  C=  0,  the  parabola,  becomes  two  real  coincident  right 
lines  ; 

when  F'z  —  B'C<iO,  the  parabola  becomes  two  imaginary  right  lines, 
as  may  be  shown  from  (s.). 

390.  Schol.  3.—  If  -  A'B  =  H-  AB  >  0. 

Cr  C' 

When  --  -  >  0,   and    —  —  <  0,  the  hyperbola  is  primary  (Art.  281). 

^1  JD 

When  C'  —  0,  the  hyperbola  becomes  two  right  lines. 

C'  C' 

When  —  —  <,   and    --  >  0,  the  hyperbola  is  conjugate  (Art.  299). 
A  B 

If  A'  —  —  B',  the  hyperbola  is  rectangular. 

391.  Examples.  —  Show  what  curves  are  represented  by  the  follow- 
ing equations. 

(1.)  £r2  +  4xy  +  f  +  3x  +  2y  +  l  =  0. 

(2.)  2x2  +  2xy  +  2y'2  =  x  —  y  —  l. 

.       (3.)  y2  +  8xy-3x  =  0. 
(4.) 


CHAPTER    IX. 
CURVES   OF  THE   THIRD   AND   FOURTH   DEGREES. 

THIRD   DEGREE. 

Proposition  1. 

392.  TJieorem.—The  general  equation  of  the  third  degree 


+ 


A, 

has  been  shown  by  Newton*  to  be  reducible  in  all  cases  to 
one  of  the  following  four  forms  : 


xy  =  x  +  £x2+Cx  +  D  ....  (c.) 
y2  =  Ax*  +  £x2+Cx  +  D  ....  (d.) 
y  =Ax3  +  Bx2  +  Cx  +  D  .  .  .  .  (e.) 

Included  under  (&.)  are  a  large  number  of  curves  of  various 
forms,  all  having  at  least  two  infinite  branches.  This  may  indeed 
be  proved  to  be  true  of  all  curves  of  odd  degree  by  Art.  215  ;  for 
a  curve  of  the  nth  degree  is  cut  in  n  points  by  one  of  the  first 
degree  —  that  is,  by  a  straight  line  ;  and  if  cut  in  an  odd  number 
of  points  by  a  straight  line,  it  cannot  be  composed  entirely  of 
finite  loops. 

*See  Newton's  "  Enumeration  of  Lines  of  the  Third  Order." 

153 


154 


CURVES  OF  THIRD  AND  FOURTH  DEGREES. 


Equation  (e.)  embraces  but  one  form,  called  the  trident,  and 
equation  (e.)  represents  one  form  only,  called  the  cubic  parabola. 

Those  included  under  (d.)  are  of  five  species,  and  are  called 
parabolas.  It  has  been  shown  that  the  shadows  of  these  five 
parabolas — that  is,  their  conical  projections — upon  planes  situated 
in  various  positions  will  give  rise  to  all  the  other  curves  represented 
by  (a.). 

Proposition  2. 
393.  TJieorem.—The  semi-cubic  Parabola 


P 

is  the  locus  of  P,  the  intersection  of  the  ordinate  QA  of 
the  parabola  if  =  4px  with  that  perpendicular  OM  to  the 
tangent  QM,  which  passes  thi*ough  the  vertex  0. 

For,  if  xl  and  yl  are  the  co- 
ordinates of  Q,  and  x'  and  y' 
of  P,  then  (Art.  128) 


is  the  perpendicular  through 
0  on  the  tangent  (Art.  249) 


and   x=xl  =  xf  .  .  .  (</.) 
is  the  line  QP.     Combining  (/.)  and  (</.),        yf  =  —  -^~  x' ; 

but         yl  =  i/4px\  =  \/4pxf ;     . ' .    y'  =  -     -#~  ~  ', 

~P 

.  • .     squaring  and  omitting  the  primes,  y2  =  — . 

394.  Exercise. — The  equations  of  the  tangent  and  normal  to  this 

-      *     ,  %/! 

( 00  —  *^i/j       £H1CL      fU        7/j  —  ( «27        OC\ ). 


curve  are,    y-y,  = 


CISSOID. 


155 


Proposition  3. 
395.  Theorem.— The  Cissoid 


is  the  locus  of  P,   the  foot  of  the  perpendicular  from  the 
vertex    0    upon  the  tangents  to  the  parabola   y2  =  —  4px. 

For,  the  equation  of  the  tangent 
to  y2  =  -4px  is  (Art.  256) 


y  =  —  mx 


m 


and  the  perpendicular  to  it  through 
0  is  (Art.  128)      y  =     , 


F      o 


X 

ra=-. 

y 


Eliminating,  m,  we  have      y2  = 


p-x 

396.  Schol. — The  polar  equation  of  the  cissoid  is 
»  sin2  0 


397.  Exercise. — (1.)    The    cissoid 

y*  =  - is  the  locus  of  P,  the 

2a  —  x 

intersection  of  A^E^  and  OB*,  when 
AiBi  and  A2J^2  are  equal  ordinates 
in  the  circle  y2  =  2ax  —  x2. 

(2.)  The  cissoid  ?/  =  — - —  is  the 
2a  —  x 

locus  of  P  when  OJ52  =  PB. 


yp 
x 


156 


CURVES  OF  THIRD  AND  FOURTH  DEGREES. 


Proposition  4. 
398.  Theorem.— The  Witch 


is  the  locus  of  P,  a  point  on  the  linear  sine  at  a  distance 
from  OX  equal  to  twice  the  linear  tatigent  of  half  the 
angle. 

For,  by  hypothesis  T 

6      _      lad  —  cos  6) 

1  +  cos  6}'        D 


y= 


.  • .     by  trig., 


399.  Exercises.  — (I.)    The     witch        ~Q 

y1  = is  the   locus   of  P  when 

2a  —  x 

A,A  :  A,B  ::OA:  A,P  =  OD. 

(2.)  The  locus  of  P,  whose  distances  plt  pz  and  pz  from  three  points 
PX,  P2  and  P3  are  such  that  either  2p1  =  p.2  +  ps,  or  p*  =  p2p$,  is  a 
curve  of  the  third  degree. 


Proposition  5. 
400.  Theorem.— The  cubic  Trisectrix* 

x*(3a  —  x} 


is  the  locus  of  P   upon  the  chord    OQ    of  the'  circle  whose 
radius  is    2a,    such  that    COQ=  QCP. 


*My  attention  was  first  called  to  the  properties  of  this  curve  as  a  trisectrix  by 
the  late  Professor  Wm.  C.  Cleveland. — AUTHOR. 


TRISECTRIX.        FOLIUM. 


157 


For,  by  geometry  the  angles  COQ 
=  CQO  and  OPC  =  PQC  +  QCP 
=  2COQ>  and  XCP  =  OPC  +  COP 
=  3COQ.  .'.  in  the  triangle  COP 

p:2a:iBm30:  sin  20, 

sin  3d 
.-.     ^  =  ^a  gin£0' 

which  is  a  polar  equation  of  the  curve. 
If    A  0  —  NQ  =  a,    then 


\T 


.-• .  p=4a  cos  0  —  a  sec  #,  which  is  another  form  of  this  polar 
equation.  By  transforming  either  of  these  polar  equations  by 
Art.  86  we  obtain  the  above  rectangular  equation. 

401.  Schol. — Let  a  line  through  (7  cut  the  curve  in  P,  P'  and  P", 
then  P'OP=P"OP'  =  60*.     For,  taking  p  in  any  position,  as  OP,  if 
XOP=0  =  Ol,  then  XCP  =30-,  and  if  a  new  position  of  ^  be  taken, 
as  OP',  such  that  0  =  0,  ±60*,  then  XCP '  =  5  (0l  ±  60°)  =30l  ±  180° 
— that  is,  CP  is  in  the  same  right  line  with  OP'.     Again,  if  another 
position  of  p  be   taken,   such   that     0  =  01±:  120°,     then   XCP"  = 
S(6l  ±  120°)  =  30l  ±  360°— i.  e.,  we  have  still  the  same  line. 

402.  Exercise. — Show  that  the  polar  equation  of  this  curve  with 
the  pole  at  0  is 

P  —  a  sec  J0, 

and  that  the  equation  of  the  curve  when  the  axes  are  the  tangents  OS 
and  027  (in  which  XOS=60*)  is 

+  y'*  =  0. 


Proposition  6. 
403.  Theorem.—  TJie  Folium 


is  the  locus  of  P  so  situated  with  reference  to  two  points 
A  and  0  that  (if  A0  =  3a)  then  ON=20M  =  2x,  and 
OQ  =  AM=  3a  +  x,  and  also  MOP  --=  POQ. 


158 


CURVES  OF  THIRD  AND  FOURTH  DEGREES. 


For,   then    ^    —  =  cos  20 ; 
3a-\-x 

.    by  trig., 


404.  Schol.  —  This  is  a  projection  of  the  trisectrix  in  which  y  =  y'~\/3. 

405.  Exercise.  —  The  equation  referred  to  the  tangents  0$and  OT 
as  axes  (in  which  XOS=45°)  is  of  the  form 

=  o. 


Proposition  7. 
406.  TJieorem.—The  Logocyclic  curve 


2a-x 
has  the  product  of  its  two  radii  vectores  constant. 

For,  transforming  to  polar  co-ordinates 
by  the  equations 

x  =  p  cos  d,    and    y  =  p  sin  0, 

transposing,   factoring   and   reducing   by 
the  relation         sin2  6  -f  cos2  0  =  1, 

On 

we  have 


cos  u 

whence       p  —  a(sec  0  ±  tan  0) ; 
.  ' .     p,p2  =  a2(sec2  6  -  tan2  0)  =  a2 ; 
.  • .     also  OA  =  a. 

This  curve  has  a  parabola  as  the  locus 
of  the  intersections  of  the  normals  drawn 
at  P!  and  P2. 


FOURTH  DEGREE. 


159 


407.  Exercise. — The  locus  of  P  when 
OP=  OB  +  OJ/is  (if  OA  =  a) 

p  =  a  (sec  0  +  cos  0), 

which  is    a   curve   of   the   third   degree. 
This  is  one  of  the  family 

p  =  a  (sec  0  +  n  cos  0), 
to  which  belong  the  cissoid  and  trisectrix. 


FOURTH    DEGREE 

408.  There  are  some  thousands  of  curves  of  the  fourth  degree, 
a  few  of  the  more  noted  of  which  are  discussed  in  the  following 
propositions. 

Proposition  8. 

409.  Theorem.—  The  Lemniscata 

or     p'p"  =  <? 

is  the  locus  of  P  moving  so  that  the  product  of  its  focal 
radii,  p'  and  p",  is  equal  to  the  square  of  half  the  dis- 
tance between  the  foci. 

Y 
For,  let  OFl=-F20  =  c, 

then  F2P  =  p"  =  [(a?  +  c)2  + 
and  F,P  =  p'  =  [(x  - 


410.  Schol.  —  If  OA  =  a,  the  equation  becomes 

(**  +  y')'  =  <*'(*'  -2/0, 

and  the  polar  equation  is 

p>  =  a'  cos  20. 

411.  Exercise.  —  Show  that  the  polar  equation  of  the  Ovals  of 

Cassini,  which  are  denned  by  the  equation 


160  CURVES  OF  THIRD  AND  FOURTH  DEGREES. 

in  which  p'  and  p"  are  focal  radii  and  b  is  any  constant,  is 


when  the  pole  is  halfway  between  the  foci,  whose  distance  apart  is  2c. 


Proposition  9. 
U2.  Theorem.— The  Limacon 


is  the  locus  of  P   at  the  intersection  of  two  lines    OP   and 
AP   moving  so  that    XAP=\XOP. 

For,  in  the  triangle  AOP, 
if    OA  =  a,   then 

p  :  a  : :  sin   |0  :  sin  ^( 


^  =^  a(^  -j-  ^  COs  0),      which  equation,  transformed  by 
Art.  86,  gives  the  above  equation. 

413.  Schol.  1. — The  equation    p  —  a  -f  ^acos  0    shows, 
since     ON=  2a  cos  tf     (Art.  197),  that  NP=a. 

414.  Schol.  2. — It  may  be  shown  in  a  manner  similar  to  that  used 
in  Art.  401  that  POP,  =  P.OP,  =  Q0\ 

415.  Exercise. — (1.)  Show  that  the  polar  equation  of  the  limacon 

with  the  pole  at  A  is  p  —  2a  cos  J0. 

(2.)  Construct  the  figure  for  the  more  general  equation  of  the  lima- 
fon,  of  which  the  foregoing  is  a  particular  case,  viz. : 


CONCHOID.        PEDAL   CURVES. 


161 


or  p  =p  cos  0  ±  a ; 

when  p^>  a  and  when  p  <ia. 


. 


Proposition  10. 
Theorem.— The  Conchoid 


is  the  locus  of  P   when    £P=±OA  =  a,    and    CO  —  b. 

For,  then  p  —  b  sec  0  ±  a,  with 
the  pole  at  C. 


Move  the  origin  to  0  by  making 

x  =  xf  +  6, 
then  a2  =  a2- 


417.  Exercise.  —  Draw  the  trifoliate  curve  p  =  a  cos  30. 

418.  Definition.  —  If  from  a  pole  0  a  perpendicular  be  let  fall  upon 
the  tangent  BP  of  any  curve  jB(7,  the  locus  of  the  foot  P  of  the  per- 
pendicular is  the  pedal  curve  with  respect  to  the  point  0  and  the  curve 
BC.     E.  G.  The  cissoid  of  Art.  395  is  the  pedal  curve  with  respect 
to  a  parabola  and  its  vertex. 


Proposition  11. 
419.  Theorem.— The  equation 

(x2  +  y2  +  x,x)2  =  aV  ± 
represents  the  pedal  curve  of  any  conic  and  a  pole  on  its 


162  CURVES  OF  THIRD  AND  FOURTH  DEGREES. 

axis,  referred  to  rectangular  axes  through  the  pole;  in 
which  xl  is  the  distance  of  the  pole  from  the  centre  of  the 
conic,  and  a2  and  ±  b2  are  the  squares  of  the  semi-axes. 

For,  the  equation  of  the  tangent  to  any  conic  is  (Art.  316) 
y  —  m(x  —  x^  +  i/cfm2  ±  b2,  where  xl  is  the  distance  of  the  origin 

x 

from  the  centre  of  the  conic,  and  y  = is  a  line  through  the 

m 

origin  perpendicular  to  this  tangent,  whence  m  —  —  -.     Combine 

«!/ 

these  equations,  and  we  obtain  the  locus  of  their  intersection, 

x2  —  xxl        Ia2x2      2 

.    .     \x    i  y       xx^ )  —  QJ  x  i 


4-20.  Schol.  1.  —  Let  the  pole  be  at  the  centre,  and  the  conic  an  equi- 
angular hyperbola  —  i.  e,,  xl  =  0,  and  a2  =  —  b2  ; 


—  that  is,  the  curve  is  the  lemniscata  (Art.  409). 

421.  Schol.  2.  —  Let  the  pole  be  at  the  vertex,  and  the  conic  a  circle 

—  i.e.,  xl  =  at  and  d2  =  b2; 

.  •  .     (x2  +  y2  ~  axj  ===  a2  (or2  +  7/2), 

which  is  a  curve  called  a  cardioid,  whose  polar  equation  is 
p  =  a  (1  +  cos  0)  =  %a  sin2|0. 

422.  Schol.  3.  —  Let  the  pole  be  at  a  distance  %a  from  the  centre, 
and  the  conic  a  circle  ; 

.  •  .     (x2  +  y2  -  Zaxf  =  a?  (x*  +  y2), 

which  is  the  equation  of  the  limacon  (Art.  412). 


.  —  The  student  whose  time  is  limited  may  complete  the  course  from  this 
point  by  taking  four  propositions  in  Chapter  XI.,  or  he  may  with  greater  advan- 
tage take  three  propositions  in  Chapter  X.,  four  or  seven  in  Chapter  XL,  and  three 
in  Chapter  XII. 


CHAPTER   X. 
HIGHER  ALGEBRAIC  CURVES. 

APPROXIMATE   CURVES. 

Proposition  1. 

423.  Theorem.  —  Parabolic  Curves  represented  by  equations 
of  the  form  y*  =  axr,  in  which  r  and  t  are  different 
positive  integers,  have  the  axis  of  x  tangent  to  them  at 
the  origin  when  r  >  t,  but  the  axis  of  y  tangent  at  the 
origin  when  r  <t. 

For,  at  any  points  (x2,  y2)  and  (x3,  y3)  upon  the  curve  we  have 

y£  —  ax2r     and     y$  —  ax3r. 
Subtract,  .  •  .     yj  -  yj  =  a(x2r  -  x3r), 

V-7/3     a(x2r~l  +'x2r~2X3  +  *2r"V  +  ----  +  a**-1). 

' 


But  (Art.  90) 


x  —  x3     x2  -  x3 
Substitute,  and  then  let  xl  =  x2  =  x3    and     y\  —  y^  =  y& 


arx 


x  —  xl       ty 


J 

t— I 


Then  equation  (a.)  represents  a  tangent  to     yi  =  axr    at    (xl}  y 
In  (a.)  let  xl=yl=0. 

Whenever  r  >  t    and  hence     T  —  1  >  t  —  1, 

then,  by  algebra,  — 4n  =  0,     . ' .     equation  (a.)  becomes  y  =0, 

y\ 

163 


164 


HIGHER  ALGEBRAIC  CURVES. 


.  ' .     (Art.  106)  this  tangent  at  the  origin  is  the  axis  of  x. 
Whenever     r  <£     and  hence     r  —  1  <£  —  1, 


then  by  algebra 


.  • .     (Art.  106)  this  tangent  at  the  origin  is  the  axis  of  y. 


r>t 


r<t 


X 


r<t 


424.  Schol.  1. — When  r  is  odd  and 
t  is  even,  the  curve  is  symmetric  about 
the  axis  of  x  (Art.  237,  1st),  and  has 
either   of  two   forms   near   the   origin 
according  as  r  >  t  or  r  <  t,  the  former 
having  a  cusp  at  the  origin. 

425.  Schol.  2. — When  r  is  odd  and 
t  is  odd,  the  curve  is  symmetric  in  oppo- 
site quadrants  (Art.  238,  2d),  and  there 
are   two   forms   according   as   r  >  t  or 
r  <  t,  both  having  a  point  of  inflexion 
at  the  origin. 


426.  Schol.  3. — When  r  is  even  and  t  is  even,  the  equation  may 
represent  at  least  two  separate  curves.  For,  the  equation,  when  both 
members  are  positive,  takes  the  form  y2H  =  6V"1. 

Transpose  and  factor,     .  * .     (yn  -f-  &cm)  (yn  —  bxm)  =  0. 
.'.     by  algebra,  y  =  bxm  and  yn  =  —  bxm. 

But  when  one  member  is  positive  and  the  other  negative,  the  curve  is 
imaginary. 

When  r  =  t,  the  equation  represents  one  or  more  right  lines. 


Proposition  2. 

'  427.  Theorem.— Hyperbolic  Curves  represented  ly  equations 
of  the  form  y*xr  =  a,  in  which  r  and  t  are  positive  in- 
tegers, all  have  both  the  axes  of  x  and  y  tangent  to  them 
at  infinity— i.  e.,  both  are  asymptotes. 


PARABOLIC  AND  HYPERBOLIC  CURVES. 


165 


For,  at  any  points  (x2)  y2)    and   (x3, 3/3)  upon  the  curve  we  have 
2/2*  =  ax2~r    and    y$  =  ax3~r. 

Subtract,    .  * .     y2*  —  yj  =  a(x^r  —  x3~r)  =  — 

t  j2 j& V    2  2 3  2 3      ' 

*^2         "^3  *^2  *^3  \2/2  2/2       2/3        ^2       2/3 

Substitute  in  the  equation  of  Art.  90,  and  then  let 

T 


y-y\ 


ar 


x—x, 


r+l 


But 


2/1  - 


^-jf (6.). 

re  —  ^         ^ 

Equation  (5.)  represents  a  tangent  to    ytxr  =  a    at    ( 
In  (6.)  let  a?!  =  oo,  then  y^  —  0  and     -  77  =  0, 
,  * .     (Art.  106)  this  tangent  at  (oo,  0)  is  the  axis  of  x. 

Again  in  (6.)  let  yl  —  oo,  then  x1  =  0  and   —  — -1  =  oo, 
.  * .     this  tangent  at  (0,  oo)  is  the  axis  of  y. 


428.  Schol.  1.— When  r  is  odd  and  t  is 

even,  the  curve  is  symmetric  about  the  axis 
of  x  (Art.  237,  1st). 


429.  Schol.  2. — When  r  is  odd  and  t  is 

odd,   the    curve    is    symmetric    in    opposite 
quadrants  (Art.  238,  2d). 


166  HIGHER  ALGEBRAIC  CURVES. 

430.  Schol.  3. — When  r  is  even  and  t  is  even,  the  equation  may 
represent  at  least  two  separate  curves  (Arts.  236,  426). 

431.  Schol.  4. — It  appears  from  the  nature  of  rectangular  co-ordi- 
nates that  when  in  the  equation  of  any  parabolic  or  hyperbolic  curve 
discussed  in  this  or  the  preceding  proposition,  y  is  written  for  x  and 
x  for  y,  the  curve  is  thereby  revolved  180°  about  the  line  whose  equa- 
tion is  x=y  — i.e.,  about  the  bisector  of  the  first  angle. 

But  if  —  y  and  —  x  be  written  for  x  and  y  respectively,  the  curve 
is  thereby  revolved  180°  about  the  line  x  —  — y  — i.e.,  the  bisector 
of  the  second  angle.  If  — y  be  written  for  y,  the  curve  is  thereby 
revolved  180°  about  the  axis  of  x ;  but  if  —  x  be  written  for  x,  the 
curve  is  revolved  180°  about  the  axis  of  y. 

Any  combination  of  these  replacements  may  be  effected  by  perform- 
ing them  successively  (cf.  Arts.  237,  238). 


^Proposition  3. 

432.  Theorem.— 1st.    If  a  given   curve  has  one  or  more 
branches  through  the  origin  (Art.  213),  either  a  parabolic 
curve  or  right  line  may  be  found,  one  for   each  branch, 
which  also  passes  through  the  origin,  and  which  in  shape 
and  direction  approximates  to  the  branch  near  the  origin. 

%d.  If  a  given  curve  has  infinite  branches,  either  a  para- 
bolic curve,  hyperbolic  curve  or  right  line  may  be  found, 
one  for  each  branch,  which  approximates  to  the  position 
and  direction  of  the  branch  at  infinity. 

This  proposition  is  one  of  the  general  results  of  succeeding 
propositions  in  this  chapter. 

433.  Schol. — By  moving  the  origin  to  different  points  of  the  given 
curve,  the  shape  and  direction  of  the  curve  may  then  be  found  by 
means  of  its  less  intricate  approximate  curves.     Hence  the  approximate 
curves  depend  on  the  position  of  the  origin. 


APPROXIMATE  CURVES. 


167 


EXPONENTIAL   POLYGON. 

434.  The  exponential  polygon  *  is  a  device  which  enables  us  to 
find  readily  which  of  the  terms  in  the  equation  of  a  curve  of 
high  degree  may  be  neglected  for  the  purpose  of  obtaining  each 
of  the  simpler  equations  which  represent  curves  approximating 
to  the  different  branches  of  the  given  curve. 

435.  Exponential  Axes.     Draw  any  two  lines    OR   and    OT  at 
right  angles  as  exponential  axes  ;  and  using  the  exponents  r  and  t 
of  any  term  ax'y*  as   the  exponential   co-ordinates,  locate  with 
respect  to    OR  and    OT  the  representative  point  (r,  t)  ;  and  in 
the  same  manner  locate  a  representative  point  for  each  term  of 
the  given  equation  by  using  its  exponents  as  co-ordinates. 

E.  G.  In  the  equation 

(y2  -  ax?  (x  -  of  -  a?xy  (x*  +  y*  -  a2)  =  0, 
or  % 


ay  —  2a?xy*  +  aV  —  aVy  —  aV?/3  +  ctxy  =  0, 


R 


the  term  a2?/4  has  the  representative  point  (0,  4)  marked  by  a  small 
circle  on  OT.     The  term  —Batfy*  has  the  representative  point  (3,  2), 

*  The  exponential  polygon  here  used  is  in  principle  the  same  as  the  "  ana- 
lytical triangle"  and  "parallelogram"  employed  by  Newton  and  others. 


168  HIGHER  ALGEBRAIC  CURVES. 

and  the  other  terms  have  their  representative  points  as  shown  by  the 
small  circles  marked  upon  the  figure. 

436.  Remark. — It  will   be  seen  that  we   have   thus    effected  an 
arrangement  of  the  terms  of  an  equation  in  rank  and  file  according  to 
the  powers  of  x  and  y  in  those  terms,  and  that  the  representative 
points  show  only  this,  viz. :  that  terms  containing  certain  powers  of 
x  and  y  occur  in  the  equation ;  they  in  no  sense  represent  points  of  the 
curve.     In  locating  the  representative  points  all   coefficients  are  dis- 
regarded, since  they  in  no  way  affect  the  position  of  those  points. 

437.  Definition. — Draw  the  smallest  convex  polygon,  having 
representative  points  at  its  corners,  which  can  contain  within  it 
or   upon    its   sides    all   the    representative    points :    this   is   the 
exponential  polygon. 

438.  Exponential    Equations. — The    sides   of  this    polygon   are 
straight  lines,  and  may  be  conveniently  represented  by  expo- 

r      t 

nential  equations  such  as  -  +  7  =  1,  which  will  facilitate  our  dis- 
cussion of  the  polygon.  The  equation  of  the  curve  is  assumed  to 
contain  no  negative  exponents,  hence  no  representative  points  are 
on  the  negative  side  of  either  of  the  axes  of  r  or  t.  There  may 
be  fractional  exponents  in  the  given  equation,  though  in  the  cases 
we  treat  we  shall  assume  the  exponents  to  be  integers. 

It  is  also  assumed  that  neither  x  nor  y  enters  every  term  of 
the  given  equation ;  for  if  every  term  contain  x,  for  instance,  the 
equation  is  exactly  divisible  by  xt  and  that  factor  may  be  removed. 
Since  those  terms  which  do  not  contain  y  have  their  repre- 
sentative points  on  OR,  and  those  which  do  not  contain  x  have 
theirs  on  OT,  it  is  evident  that  the  polygon  has  either  a  corner 
or  a  side  in  each  of  the  axes  of  r  and  t. 

439.  Sides. — If  each  side  of  the  exponential  polygon  be  pro- 
duced indefinitely,  five  kinds  of  sides  may  be  distinguished,  as 
follows : 

1st.  Sides  whose  intercepts  on  the  axes  of  r  and  t  are  both 
positive,  and  which  also  lie  between  the  origin  and  the  rest  of 
the  polygon. 


APPROXIMATE  EQUATIONS.  169 

2d.  Sides  which  have  both  intercepts  positive,  and  which  have 
the  rest  of  the  polygon  between  themselves  and  the  origin. 

3d.  Sides  whose  intercepts  are  one  positive  and  one  negative ; 
and  also  sides  through  the  origin,  all  parallels  to  which  have  inter- 
cepts, one  positive  and  one  negative. 

4th.  Sides  that  coincide  with  either  of  the  axes. 

5th.  Sides  parallel  to  either  of  the  axes,  which  have  the  rest  of 
the  polygon  between  themselves  and  the  origin. 

440.  N.B. — If  now   from    the   given   equation   all   terms   be 
omitted   except   those   whose    representative    points   lie   upon  a 
single  side  of  the  exponential  polygon,  let  us  call  the  result  "  the 
approximate  equation  fop  that  side."     And  if  from  any  equation 
all  terms  be  omitted  except  those  whose  representative  points  lie 
on  any  assumed  right  line,  let  us  call  the  result  "  the  approximate 
equation  for  that  line." 

441.  The  following  proposition  will  be  shown  to  be  another  of 
the  general  results  of  the  succeeding  propositions  of  this  chapter. 

Proposition  4. 

Theorem. — Each  approximate  equation  for  a  side  of  the 
exponential  polygon  represents  an  approximate  curve. 

When  the  approximate  equation  is  for  a  side  of  the  first  kind, 
the  approximation  is  near  (0,  0). 

For  a  side  of  the  second  kind,  the  approximation  is  near  (oo,  oc). 

For  a  side  of  the  third  kind,  the  approximation  is  near  (0,  oo) 
or  (oo,  0). 

For  a  side  of  the  fourth  kind,  the  approximation  is  an  inter- 
section of  the  curve  with  the  axis  of  x  or  y. 

For  a  side  of  the  fifth  kind,  the  approximation  is  at  infinity, 
and  to  a  right  line  parallel  to  the  axis  of  x  or  y. 

Or,  as  it  may  be  stated  in  other  words,  each  approximate 
equation  for  a  side  of  the  exponential  polygon  represents  an 


170  HIGHER  ALGEBRAIC  CURVES. 

approximate  curve,  when  x  is  made  infinite  in  all  approximate 
equations  for  right-hand  sides,  and  infinitesimal  in  those  for  all 
left-hand  sides  ;  while  at  the  same  time  y  is  made  infinite  in  all 
approximate  equations  for  upper  sides,  and  infinitesimal  in  those 
for  all  lower  sides. 

442.  E.  G.  —  In  the  example,  Art.  435,  for  sides  of  the  first  kind  the 
equations  are  (see  figure) 

a\f  +  cfxy  =  0,    or    y*  —  —  a?x, 
and  a*x2  +  cfxy  =  0,    or    y  =  —  x, 

which  represent,  as  will  be  shown,  curves  approximating  to  the  branches 
of  the  given  curve  near  the  origin. 

There  is  one  side  of  the  second  kind  from  which 
#y  —  2ax\f  +  aV  =  0, 

whence  y2  =  ax,  which  represents  a  curve  which  approximates  to  one 
of  the  infinite  branches  of  the  given  curve. 

There  is  one  side  of  the  fourth  kind  from  which 

aV  =  0, 


whence  x  =  a,  which  is  the  intersection  of  the  curve  with  the  axis  of  x. 
There  is  one  side  of  the  fifth  kind  from  which 

«y  —  2axy*  +  #y  =  0, 

whence    x  =  a,  which  is  the  equation  of  a  straight  line  which  approxi- 
mates to  one  branch  of  the  curve  near  (a,  oo). 


Proposition  5. 

443.  TJieorem.—TIxe  approximate  equation  for  any  assumed 
right  line  is  of  the  form 


(r  and  t  bei7^g  positive  integers),  whenever  the  intercepts 
on  that  line  upon  the  exponential  axes  are  both  positive; 
but  is  of  the  form 

y  V  —  a 

ivhenever  they  are  one  positive  and  the  other  negative. 


POSITIVE  AND  NEGATIVE  INTERCEPTS.  171 

T  t 

For,   let    --  \-~  -=1   be   the   equation  of  the   assumed  line 

ro      co 
referred  to  the  axes  of  r  and  t,  in  which  r0  and  t0  are  the  inter- 

cepts. If  rl  and  £j  are  the  exponents  of  one  term  of  the  approxi- 
mate equation  for  this  line,  r2  and  t2  those  for  another,  r3  and  £3 
those  for  another,  etc.,  then  we  have  the  equations, 

Tl       ,        ^1  y  r2       ,        ^2  y  r3        ,        ^3  y 

~  +  T~  —  1,       7  +  7~  —  A       r  ""  T~  =  • 

ro        ^o  ro        Jo  ro        co 

Subtracting,  transposing,  etc., 


which  is  the  relation  between  the  exponents  of  the  terms  whose 
representative  points  lie  on  this  line  —  i.  e.,  the  relation  of  the 
exponents  in  the  general  form  of  the  approximate  equation  for 
this  line. 

This  general  form  may  be  written 

aXY  l  +  a^Y2  +  a3^Y  3  +  •  •  •  =0  .  .  .  (6.). 
Divide  this  equation  through  by  a^y*1, 

4-  .  .  .  =0, 


or  a,     a2-y  a 

which  by  equation  (a.)  becomes 


_ 

from  which,  by  algebra,  one  or  more   constant  values  of  x  <0y 

_rp 
can  be  obtained.     In  case  r0  and  £0  are  both  positive,  x  toy  =  c 

becomes     yto  —  c<0of°,  and  is  of  the  form  yi  =  axr  .  .  .  (e.). 

But  in  case  one  intercept,  as  r0,  is  negative, 

f£j 
let  TO  =  —  TO'}  then  xioy  =  c,  or  ytoxr°r  =  cto, 


172  HIGHER  ALGEBRAIC  CURVES. 

which  is  of  the  form  ytxr  =  a  .  .  .  (d.).  We  shall  call  the  reduced 
equations  (c.)  and  (d.)  "the  approximate  equations,"  in  distinction 
from  (b.)  "  the  general  form  of  the  approximate  equation  for  any 
assumed  line." 

444.  ScJtol. — The  approximate  equations  (c.)  and  (d.)  for  parallel 
lines  can  differ  only  in  the  value  of  the  coefficient  a. 

_ro 

For,  in  the  equation  x  TQy  =  c,  the,  exponent,  which  is  the  negative 
ratio  of  the  exponential  intercepts,  has,  by  similarity  of  triangles,  the 
same  value  for  all  parallel  lines.  And  conversely,  all  approximate 
equations  that  differ  only  in  the  value  of  the  constant  are  for  lines 
which  are  parallel.  For,  since  the  ratio  of  the  exponential  intercepts 
is  the  same  (by  similar  triangles),  the  lines  are  parallel. 

445.  E.  G.  In  the  example,  Art.  435,  the  general  form  of  the  ap- 

T         t 

proximate  equation  for  the  assumed  line        -  +  —  =  1 

4     4 

is  oV  -  aVy  +  kttff  -  a V  +  ay  =  0, 

or  x*  —  x3y  -\-  4%2y2  — 

which  is  symmetrical  in  x  and  y, 

.'.     by  algebra     -  =  -     and  ."•• . 

y    x  y 

But  -  =  ±  1  does  not  satisfy  the  equation,  therefore  all  values  of  — 

y  y 

are  imaginary. 

T         t 

For  the  parallel  to  the  assumed  line,   --\--  =  lt  the  approximate 

o      3 

equation  is  —  #aV  —  2a3xy*  =  0,     .  • ,     x  —  y  \/  —  1. 

T          t 

And  for  the  other  parallel,  -  +  -  =  1,  we  also  obtain  x  =  y  |/  —  1. 

o      o 

T  t 

For  the  line  -  -\ =  1,  the  equation  is 

<v'         & 

aV  —  aVy  =  0,     .  • .     xy  =  a*\ 
and  for  the  parallel  line          r  =  t         the  equation  is 

4a?xy  +  ctxy  =  0,     .  • .     xy  =  0    and     xy  =  ——. 

4 


RELATIVE  DEGREE  OF  TERMS.  173 


For  the  line  -  +  1  =  1, 


and  for  the  parallel  +  --  =  .?,     aty 

For  the  line  -  +  -  =  -?,       #2  =  -ay  ; 

f 

9'          ?! 

and  for  the  parallel         —  f-  -7  —  7,       #2  =  —  2ay  ; 


T         t 

and  for  the  parallel         — \-  -  =1)      x2  =  —  ~^ay- 

<g 


Proposition  6. 

446.  Theorem.  —  //z,  order  to  compare  the  degrees  of  the 
several  terms  in  the  equation  of  any  curve,  replace  x  or  y 
by  its  value  obtained  from  an  approximate  equation,  for 
any  assumed,  line,  of  the  form 

?/'  =  axr,  then 

1st.  All  terms  whose  representative  points  are  on  the  as- 

sumed line  become  of  the  same  degree. 
2d.  All  the  terms  whose  representative  points  are  on  any 

one  line  parallel  to  the  assumed  line  become  of  the  same 

degree. 
3d.  All  terms  whose  representative  points  lie  between  the 

line  and  the  origin  become  of  less  degree  than  those  whose 

representative  points  are  on  the  line,  but  all  terms  ivhose 

representative  points  are  on  the  opposite  side  of  the   line 

become  of  greater  degree. 

For,  first,  if  in  equation  (b.)  (Art.  443),  we  replace  y  by  its 

12. 

value  cxto,  then  equation  (ft.)  is  reduced  to  the  form 


,     n+-t9<1     .          .     r2+lQ<3    i         /    '3+  —  '3    , 

a/z      t0    +  a.2'x      t0    +  a3'x      *    +  .  .  .  =  0  .  .  .  (e.), 

the  degrees  of  whose  successive  terms,  according  to  equation  (a.), 
exceed  that  of  the  first  term  by 

r,-^  +-&-*i)  =^i       r3  ~ri  +       fe-*i)  =°>  etc- 


174 


HIGHER  ALGEBRAIC  CURVES. 


And,  second,  since  by  Art.  444  all  parallel  lines  have  approxi- 
mate equations  which  differ  only  in  the  value  of  the  coefficient 
a,  it  is  therefore  evident  that  the  expression  (e.)  above  differs 
from  that  which  we  should  get  by  replacing  y  with  some  value 

ro 

cV°  obtained  from  some  parallel  line — only  in  its  coefficients. 

But,  third,  since  all  the  terms  along  any  one  of  the  parallel 
lines  are  of  the  same  degree  in  x  after  the  replacement,  all  the 
terms  along  each  of  the  lines  are  of  the  same  degrees  respectively 
as  the  terms  situated  at  the  intersection  of  each  of  the  lines  with 
the  axis  of  r.  But  the  terms  along  r  increase  uniformly  in 
degree  from  the  origin ;  therefore  each  successive  parallel  count- 
ing from  the  origin  has  its  terms  of  higher  degree  than  the 
preceding. 

M7.  Schol. — This  may  appear  more  clearly  if  the  facts  be  indicated 
upon  a  diagram.  Let  us  write  the  literal  part  of  every  term  in  a 


xy 


1  x 


4 


— 

x10 

xn 

xs 

x» 

xw 

x6 

x~ 

Xs 

x» 

5 

^c 

x° 

a-7 

Xs 

x~ 

a8" 

> 

<, 

x° 

X7 

1 

X 

x~ 

a:3 

x^ 

^L  x* 

2       4       6\/ 

general  equation  of  the  sixth  degree  just  at  the  right  and  above  its 

representative  point.     Suppose  the  assumed  line  to  be  -  +  -  —  1; 

6     3 


.'.     (equation  (c.)),     y  —  ax1, 


for, 


0  =  6    and    t0—3, 


and  the  terms  thus  become  by  the  replacement  of  y  as  seen  in  the  right 
hand  figure. 


RELATIVE  DEGREE  OF  TERMS.  175 

And  the  same  result  would  have  appeared  had  we  made  use  of  any 
line  the  ratio  of  whose  intercepts  is  the  same. 

448.  Cor. — When  one  intercept  is  infinite  (as  £0),  we  have 
When  TQ  —  IQ,    then  y  =  bx. 


^Proposition  7. 

449.  Tlieorem.—In  order  further  to  compare  the  degrees 
of  the  several  terms  in  the  equation  of  any  curve,  replace 
y  by  its  value  obtained  from  the  approximate  equation  for 
any  assumed  line  of  tine  form  ytxr  =  a  (the  intercept  on 
the  axis  of  t  being  negative,  and  that  on  r  positive)  ;  then, 


1st.  All  terms  whose  representative  points  are  on  the 
assumed  line  become  of  the  same  degree. 

2d.  All  the  terms  whose  representative  points  are  on  any 
one  line  parallel  to  the  assumed  line  become  of  the  same 
degree. 

3d.  All  terms  whose  representative  points  lie  between  the 
line  and  the  origin  become  of  less  degree  than  those  whose 
representative  points  are  on  the  line,  but  all  terms  whose 
representative  points  are  on  the  opposite  side  of  the  line 
become  of  gr  'eater  degree. 

4th.  If  x  be  replaced  instead  of  y,  the  terms  whose 
representative  points  lie  on  the  side  toward  the  origin  are 
of  greater  degree,  and  those  on  the  opposite  side  of  less 
degree. 

5th.  A  corresponding  statement  is  true  when  the  intercept 
on  the  axis  of  r  is  negative,  and  that  on  t  positive. 

For,  the  first,  second  and  third  parts  of  the  proposition  are 

proved  in  the  same  manner  as  in  Art.  446  ;   and  fourth,  since 

i_  ^ 

ytxr  =  a,     .'.     x  —  (ay~t)r,     and    y  =  (ax~r}t  —  i.e.,    the   greater 
powers  of  x  are  the  less  powers  of  y,  and  vice  versa. 


176 


HIGHER  ALGEBRAIC  CURVES. 


450.  Schol. — This  proposition 
may  be  illustrated  as  was  the  pre- 
vious one.  Assume  the  line 


451.  Cor. — When    one    inter- 
cept is  infinite  (as  £0),  we  have 


0 


X 


xl 


*1 


I 


2.  Exercise.  —  Write    the 
literal  parts  of  the  terms  in  a  similar  manner  when  x  is  replaced. 

Proposition  8. 
453.  Theorem.— An  approximate  equation  of  the  form 

y*  =  axr 

for  a  side  of  the  exponential  polygon  of  the  first  kind  (Art. 
439)  represents  a  curve  which  approximates  to  the  original 
curve  for  infinitesimal  values  of  x  and  y  (i.  e.,  near  the 
origin) ;  and  an  approximate  equation  for  a  side  of  the 
second  kind  is  of  the  same  form,  but  represents  a  curve 
that  approximates  to  the  original  curve  for  infinite  values 
of  x  and  y. 

For,  if  the  general  form  of  the  approximate  equation  for  a  side 
of  the  first  kind  be  obtained  in  the  manner  shown  in  Art.  443, 
equation  (6.),  and  the  value  of  y  obtained  from  this  equation  be 
substituted  in  every  term  of  the  equation  of  the  curve,  it  has 
been  shown  (Art.  446)  that  the  terms  whose  representative  points 
lie  on  this  side  are  then  of  lower  degree  than  the  rest ;  for,  this 
side  lies  between  all  others  and  the  origin.  When  x  is  made  infini- 
tesimal after  the  replacement  (i.  e.,  both  x  and  y  infinitesimal  in 
the  original  equation),  all  other  terms  become  vanishing  quantities 
in  comparison  with  those  whose  representative  points  lie  on  this 
side.  In  the  same  way  it  is  evident  that  the  terms  whose  repre- 
sentative points  are  on  a  side  of  the  second  kind  are  of  higher 
degree  than  the  rest,  and  that  all  other  terms  are  vanishing 
quantities,  when  infinite  values  are  assumed  for  x  and  y. 


APPROXIMATE  CURVES.  177 

454.  Examples.  —  Show  that  we  have  the  following  approximate 
curves. 


(1.)  In  the  Cissoid,     x3=py'i, 

(2.)  In  the  Witch,    f  =  2ax,  x  —  2a. 

(3.)  In  the  Cubic  Trisectrix,    y=±.x 

(4.)  In  the  Folium,    y  =  ±x,  x  =  —  a. 

(5.)  In  the  Lemniscata,  and  the  equilateral  hyperbola,  x  =  ±y. 

(6.)  In  the  Limacon,    y=±x  -\/~3, 

(7.)  In  the  Cardioid,    x3  =  —  \  ay\ 

Proposition  9. 
455.  Tlieorem.—tAn  approximate  equation  of  the  form 


for  a  side  of  the  third  kind  represents  a  curve  that  approxi- 
mates to  the  original  curve  for  infinite  values  of  x  and 
infinitesimal  values  of  y,  when  its  intercept  on  the  axis 
of  r  is  positive,  and  that  on  t  negative  (that  is,  for  that 
branch  of  the  hyperbolic  approximate  curve  to  which  the 
axis  of  x  is  an  asymptote)  ;  but  it  represents  a  curve  that 
approximates  to  the  original  curve  for  infinite  values  of 
y  and  infinitesimal  values  of  x,  when  its  intercept  on 
the  axis  of  t  is  positive,  and  that  on  r  is  negative. 

For,  the  terms  whose  representative  points  lie  on  a  side  of  the 
third  kind  are  of  higher  degree  in  x  and  lower  in  y  (or  vice 
versa)  than  any  other  terms  in  the  equation  (Art.  449)  ;  there- 
fore, if  x  =  co  nearly,  and  y  —  0  nearly,  all  other  terms  are  vanish- 
ing quantities. 

456.  Schol.  1.  —  The  general  form  of  the  approximate  equation  for 
a  side  of  the  fourth  kind  contains  x  only,  or  else  y  only.  Suppose  it 
is  x  only,  then  the  other  terms  of  the  original  equation  vanish  when 
y  =  0.  Hence  we  obtain  the  points  of  intersection  of  the  curve  with 
the  axis  of  x. 

12 


178  HIGHER  ALGEBRAIC  CURVES. 

457.  Schol.  2. — The  general  form  of  the  approximate  equation  for 
a  side  of  the  fifth  kind  contains  the  same  powers  of  either  x  or  y, 
according  as  the  side  is  parallel  either  to  the  axis  of  r  or  t.  Suppose 
each  term  has  in  it  the  same  power  of  x,  and  on  dividing  through  by 
that  factor  the  equation  will  consist  only  of  powers  of  y  (i.  e.,  y  =  a 
by  Art.  451) ;  then  all  the  terms  of  the  original  equation  whose  repre- 
sentative points  do  not  lie  on  this  side  vanish  in  comparison  on  making 
y=a  and  x  =  oo  nearly  (i.e.,  the  curve  approximates  at  infinity  to 
the  line  y  =  a).  A  similar  statement  is  true  when  x  =  &  constant. 


Proposition  10. 

458.  Theorem.— 1st.  When  an  equation  has  no  constant 
term — i.  e.,no  representative  point  at  (0,  0)—the  curve  has 
a  single  branch  through  the  origin— i.  e.,  it  has  a  single 
point  at  (0,  0). 

'^2d.  When  an  equation  has   no  constant  term  nor  terms 
of  the-  first  degree— i.  e.,  no  representative  points  at    (0,  0), 
^(<?j"^)    and    (I,  0^^-then  the  curve  has  two  branches  through 
the  origin — i.e.,  it  has  a  double  point  at    (0,  0). 

3d.  When  in  addition  it  has  no  terms  of  the  second  de- 
gree, it  has  a  triple  point  at  the  origin,  etc.,  etc. 

For,  first,  if  there  is  no  constant  term  in  the  equation,  but 
there  is  a  term  of  the  first  degree  (containing  y,  say),  then  there 
is  one,  and  but  one,  approximate  equation  of  the  form  y  =  axr,  as 
may  be  seen  by  finding  the  representative  points  of  any  such 
equation. 

And,  second,  if  there  is  a  term  of  the  second  degree,  but  no' 
constant  term,  and  no  terms  of  the  first  degree,  it  will  appear 
from  consideration  of  the  exponential  polygon  either  that  there 
are  two  sides  of  the  first  kind,  in  which  case  the  proposition  must 
be  true,  or  that  in  the  approximate  equation  yt  =  axr,  r  =  2,  or 
t  =  2.  Suppose  t=2,  then  y  =  ±  \/^axr ;  hence  as  x  decreases  to 
zero,  there  are  two  infinitesimal  values  of  y,  which  vanish  to- 
gether. (It  will  be  seen  that  a  cusp  is  a  double  point  as  well  as 
the  intersection  of  two  branches  which  cross  at  an  angle.) 


MULTIPLE  POINTS.  179 


Also,  third,  from  similar  considerations  the  proposition  may  be 
extended  to  the  case  of  a  triple  point,  etc.,  etc. 

459.  Schol. — If  the  origin  be  moved  to  any  point  of  a  curve,  then 
an  approximate  equation  for  a  side  of  the  first  kind  may  be  found  at 
that  point  which  will  show  the  direction  of  the  curve  at  that  point, 
and  the  kind  of  singularity,  if  any  exists  at  that  point.  In  general, 
the  approximate  equations  are  changed  by  transformation  of  co-ordi- 
nates. Review,  Art.  432. 

4-60.  Examples. — Discuss  the  following  curves  with  reference  to 
their  multiple  points  at  the  origin,  their  approximate  curves,  and  their 
intersections  with  the  axes  of  x  and  y. 

(1.) 
(2.) 

(3.) 

(4. )  *y  -  Saxy  -  V( 

(5.) 

(6.) 

©i»+i     /i/^n+l 
4-  (f )       =  1,  in  which  n  is  an  integer. 
W 

(8.)  (-}    +(7)    =1,  when  n<l,  and  when  rc>  1,  but  not 

W        W  * 

necessarily  an  integer.     (If  n  =  ^  the  curve  is  a  parabola.) 

(9.)  Discuss  the  double  points  of  the  curves  in  Chapter  IX. 

401.  Exercise. — Move  the  origin  in  the  example,  Art.  435,  to  the 
point  (a,  0),  and  show  that  the  transformed  equation  then  has  at  (0,  0) 
the  approximate  equations  x  =  %,  and  y2  =  —  2ax,  and  at  (0,  oo)  the 
approximate  equation  xly  =  a3. 

462.  General  Scholium. — The  properties  of  the  exponential  poly- 
gon may  also  be  used  to  discover  the  equation  which  will  represent 
the  general  contour  of  a  given  curve  whose  equation  is  unknown. 


CHAPTER   XI. 
TRANSCENDENTAL  CURVES. 

463.  Transcendental  Curves  are  those  which  require  the  use  of 
transcendants,  such  as  sin,  tan,  log,  etc.,  to  express  the  relation 
between  the  x  and  y  of  any  point. 

Several  of  these  are  discussed  in  the  following  propositions. 

Proposition  1. 
464."  Theorem.— The  equations 

x  —  r  (<p  —  sin  <f>)    and    y  =  r  (1  —  cos  <p) 

represent  the  common  cycloid,  which  is  the  locus  of  a  point 
on  the  circumference  of  a  circle  which  rolls  along  a  straight 
line ;  in  which  r  is  the  radius  of  the  circle,  and  <p  is  the 
angle  of  rotation. 

Y 


For,     since  y  = 
and  r  =  PC=MC, 

then     OM=  arc  MP  =  r<p  ; 

and  if  P  is  the  point  originally 
in  contact  with  0, 

then  x  =  OH  —  BM  =  r<p  —  r  sin  <p  =r 

and 


B     M 


X 


sn 


465.  Schol.  1.  —  These  equations  are  analogous  to  the  equations  of 
the  ellipse  and  hyperbola  in  terms  of  the  eccentric  angle,  and  <p  is 
called  the  auxiliary  angle  (Arts.  344,  349). 


466.  Schol.  2. — As  the  circle  continues  rolling,  an  infinite  number 
180 


PROLATE  AND   CURTATE  CYCLOIDS. 


181 


of  similar  branches  may  be  formed  of  which  the  equation  of  the  n* 
branch  is 

x  =  r  \%m:  -f  <P  —  sin  (2m:  -f  £>)]  —  r  (2mz  -f  <p  —  sin  <p) 

y  —  r  [1  —  cos  (2m:  +  ?)]=r(l  —  cos  <p). 

467.  Exercise.  —  Show  that  the  first  of  the  equations  of  Art.  464 
may  be  written 


x  =  r  cos 


or 


Proposition  2. 

468.  Theorem.— The  equations 

x  =  r  ((f>  —  m  sin  ^>),   and  y=r(l—m  cos  (p) 

represent  the  locus  of  a  point  on  the  radius  of  a  circle 
which  rolls  along  a  straight  line ;  in  which  mr  is  the  dis- 
tance of  the  point  fro?n  the  centre  of  the  circle,  and  the 
curve  is  a  prolate  or  curtate  cycloid  according  as  m<l  or 
m  >  1.  The  curve  is  also  called  a  trochoid. 


For,  since   <p  =  PCM,   mr  =  PC    and    r  =  PlC=MC, 
then  OM=  arc  MPl  —  r<p, 

then  x  =  OH—  BM=  r<p  —  mr  sin  <p  —  r(<p  —  m  sin  <p), 
and  y  =  MG—  DC=r  —  mr  cos  <p  —  r(l  —  m  cos  <p). 

The  scholia  of  Proposition  1  apply  to  this  proposition  also. 


182 


TRANSCENDENTAL   CURVES. 


469.  Exercise. — Draw  the   companion   to   the   cycloid   which  is 
determined  by  the  equations       x  =  ry>     and    y  =  r  (1  —  cos  <f>}. 


470. 


^Proposition  3. 

TJieorem.—The  equations 

(  \  /ri  +  r2 

X  ~  (ri  ~*~  r2J    COS  (f~T2  COS  ( ^ 

\          2 

V  —  ^i  ~T~  T2j  Sin  ^        7*2  SI 


represent  cun  epicycloid,  which  is  the  locus  of  a  point  on  the 
circumference  of  a  circle  which  rolls  around  on  the  outside 
of  a  fixed  circle;  in  which  TI  is  the  radius  of  the  fixed 
and  r-i  that  of  the  rolling  circle  ,  and  y  its  angle  of 
revolution. 


For,  let  OB  =  rl,  the 
radius  of  the  fixed 
circle,  and  CB  =  CP 
=  r2,  the  radius  of  the 
rolling  circle,  when  P 
is  the  point  which  was 
originally  in  contact 
with  A.  Let  the  angle 
of  revolution  AOC=  <p, 
and  the  angle  of  rota- 
tion OCP  =  (f>', 

I  A 

then     x  =  ON+  NM=  (rx  +  r2)  cos  tp  +  rz  sin  I  <p  +  <p'  —  ^  ), 

and       y  =  NC—DC=  (^  +  r2)  sin  <p  —  r2  cos  (  <p  +  <p'  —  -  J. 

.  •  .     x  =  (rj  +  r2)  cos  <p—T2  cos  (<f,  +  y'\ 
y  =  (rl  +  r2)  sin  <f>  —  r2  sin  (up  +  <pr).  " 
But,  since  rl<p=r2<p'=A£, 

eliminate  <p'}  and  we  have  the  equations  given  above. 


EPICYCLOID  AND  HYPOCYCLOID.  183 

471.  Sc/tol.  —  When  rl=rt  =  |a,  the  equations  become 

X  1  1 

-  =  cos  <P  —  —  cos  2<p  =  cos  <p(l  —  cos  <p)  +  -^ 
a 

-  =  sin  </>  —  |  sin  2<p  =  sin  <p(l  —  cos  y). 

i  ,  x      1      x' 

Let  *=,«.-*<    or   j-j—, 

.*.     --  =  cosy(l  — 
a 


and  a;'2  +  y2  —  ax'  —  a?(l  —  cos  <f>)2  +  a2  cos  ^(1  —  cos  ^)  =  a2(^  —  cos  ^), 


which  is  the  equation  of  the  cardioid  (Art.  421). 
472.  Exercise.  —  Show  that  the  equations 


=  (rljr  r2)  cos  <f>  —  mr2  cos  I  —   —  V) 

=  (n  +  ra)  sin  p  —  mr2  sin  (-  —  -^  J 

V    r        / 


represent  the  prolate  and  curtate  epicycloid  —  i.  e.,  the  epitrochoid. 


^Proposition  4. 
473.  TJieorem.—The  equations 


=  (rl—  r2)  coa  <p  +  r2  cos 


=      ~  r    sn     -  r  sn 


I  J    —  2  ^>  1 


represent  an  hypocycloid,  which  is  the  locus  of  a  point  on 
the  circumference  of  a  circle  which  rolls  around  the  inside 
of  a  fixed  circle. 

For,  these  equations  may  be  obtained  by  a  method  similar  to 
that  used  in  the  previous  proposition,  or  by  putting  —  r2  for  +  r2. 


184  TRANSCENDENTAL   CURVES. 

474.  Schol.  1.  —  When  rt  =  #r2,  the  equations  become 

x  =  2r.i  cos  <f>,   and   y  =-0,  in  which  x  may  have  any  value  between  -f  rl 
and  —  r1}  and  the  curve  is  reduced  to  the  diameter  of  the  fixed  circle. 

475.  Schol.  2.  —  When  ^  =  ^r2,  we  have 

x  =  3r.2  cos  <p  +  r2  cos  <%  =  ^r2  cos3  $P 


one  of  the  curves  of  the  form  given  in  Art.  460,  Ex.  (8). 

y         77 

It  may  be  shown  that  the  line  —  \-  -  =  1  is  always  tangent  to  it  when 

a      o 

a?  -f-  H1  =  fj*  —  that  is,  on  condition  that  this  tangent  line  has  a  length 
=  rj  intercepted  between  the  axes. 

476.  Exercise.  —  Show  that  the  equations 

x  =  (rl  —  r2)  cos  <p  +  mr2  cos  (—  -  -2 

\     r2 

y  —  (T!  —  r2)  sin  <p  —  mr2  sin  I-  -  -  <p  ) 

V    ^'2        ' 

represent  the  prolate  and  curtate  hypocycloids  —  i.  e.,  the  hypotrochoid 
—  and  also  that  when  rx  =.2rt  the  hypotrochoid  becomes  the  ellipse 

,          f          _2 
(1  +  mfr?      (1-  mjr? 


^Proposition  5. 
477.  Tlieorem.—The  equations 

x=r  (cos  <p  4-  <p  sin  y>)     and     y  —  r  (sin  ^  —  (p  cos  $p) 

represent  the  involute*  of  a  circle,  which  is  the  locus  of  P 
at  the  end  of  a  thread  unwound  froin  the  circumference  of 
a  circle  whose  radius  is  r. 

*  This  involute  is  a  spiral  which  has  no  real  points  within  the  generating  circle. 


REPEATING  CURVES. 


185 


For,  if 

then      OBN^-  BPD  =  ^-v, 
& 

and          BP  =  AB=--ry, 

...  x  =  OM=ON+DP, 

and  y  =  MP  =  NB-DB; 

.  ' .  x  —  r  cos  <p  +  ry>  sin  <p, 

and  y  =  r  sin  ^  —  r^>  cos    ^. 

478.  Exercise. — Show  that  the  polar  equation  of  this  curve  is 


or 


TRIGONOMETRIC  LOCI. 

Proposition  6. 

479.  Theorem.— The  equations 

y  =  sin  x,     y  =  tan  x,    y  =  sec  x, 
represent  periodic  or  repeating  curves. 

For,,  let  the  curves  be  constructed  by  tracing 
them  through  points  determined  by  tables  of 
natural  sines,  tangents  or  secants,  or  by  drawing 
the  lines  as  in  Fig.  1.  We  then  have  the  curve  of 

"X  Fig  2. 


Figl. 


186 


TRANSCENDENTAL   CURVES. 


sines,  Fig.  2;  the  curve  of  tangents,  Fig.  3;  and  the  curve  of 
secants,  Fig.  4. 

Since  sin  x  =  sin  (2nn  +  x), 

and  tan  x  =  tan  (2nn  +  x), 

and  sec  x  =  sec  (2nx  +  x) 

(when  n  is  an  integer),  the  curves  repeat  themselves  along  x. 


Fig3. 


X 


Fig  4. 


X 


480.  Schol.  1. — The  equation 

xs       x> 
y  =  ^x  =  x-  —  +  —  -  etc.,  by  trig., 

.  * .  the  right  line  y  =  x  approximates  to  the  curve  at  the  origin 
(Art.  453) — i.  e.,  is  tangent  to  it.  Move  the  origin  to  (^TT,  Ij, 
.-.  x  =  af  +  jji:  and  y  =--  y'  +  1,  .'.  tf  +  1  =  sin(#'  +  |TT) 

£'2  yft 

=  'coBxf  =  l —+— —etc.,  by  trig.,     .'.     the  parabola  xn  =  —  2tf 

&        <£Jf        f 

approximates  to  the  curve  at  (^r,  1\.  .'.  also  y  ='cos  ^  is  the  same 
curve  as  y  =  sin  x  with  a  different  origin,  as  is  also  y  =  versin  #. 


TRIGONOMETRIC  LOCI.  187 

481.  Schol.  2.—  The  equation 

x?      2x& 

y  =  i&nx  =  x  +  —  -}-  —  +  etc.,  by  trig., 
o        lo 

.  •  .     y  =  x  is  tangent  at  the  origin.     This  may  be  shown  to  be  the 
same  curve  as  y  =  cot  x  with  a  different  origin. 

482.  Schol.  3.—  The  equation 


.  • .    y  = ,  =  1  H h  etc.,  by  division, 

1  —  ^x1  +  etc. 

.  • .  at  (0,  1)  the  parabola  x2  —  2y  approximates  to  the  curve.  The 
curve  y  =  cosec  x  differs  from  y  —  sec  x  only  in  the  position  of  the 
origin. 

483.  Exercise. — Construct  the  curve  x  =  log  y    or    y  =  of  when 


Proposition  7. 

484.  TJieorem.—If  in  the  equation  representing  any  locus 
(say,  <p(x,  y)=0*),  any  trigonometric  function  of  y  be 
written  in  place  of  x,  and  also  any  trigonometric  func- 
tion of  x  in  place  of  y,  the  equation  thus  obtained  (say, 
tp  (sin  y,  tan  x)  =  0)  represents  a  trigonometric  locus  which 
may  be  readily  traced  from  its  relation  to  the  original 
curve  (viz.,  </>(x,  y)  —  0). 

The  truth  of  this  proposition  will  appear  as  the  result  of  several 
succeeding  propositions.  The  nature  of  the  substitutions  made 
may  be  seen  from  the  following  equations.  From  the  equation 
of  the  right  line  y  =  mx  we  thus  obtain, 

*This  expression  is  read  "0  function  of  x  and  y  equal  to  zero." 


188 


TRANSCENDENTAL   CURVES. 


sin  x  =  m  sin  y,  tan  x  =  m  sin  y, 

sin  x  —  m  cos  y,  tan  x  —  m  tan  y, 

sin  a;  =  m  tan  y,  tan  x  =  m  sec  y, 

sin  re  =  m  sec  y,  tan  x  —  m  cos  y, 


etc.,  etc. 


etc.,  etc. 


From  the  equation  of  the  parabola  y2  =  mx  we  thus  obtain 
sin2  re  =  m  sin  y,         cos2#  =  m  tan  y, 
tan2#  =  m  cos  y,         sec2#  =  m  versin  y, 
etc.  etc. 

Proposition  8. 

485.  Theorem.— If  the  locus  of  the  point  P,  which  moves 
according  to  some  law  expressed  by  the  equation  <p  (#,  y)  =  0, 
be  traced,  together  with  the  auxiliary  curves  represented  by 
the  equations  x  =  siny  and  y  =  sin  x,  then  the  trigono- 
metric locus  whose  equation  is  ^(sin  y,  sin  x)  =  0  is  the  locus 
of  the  point  P'  situated  at  the  corner  of  the  rectangle 
SPTP',  whose  sides  are  parallel  to  the  axes  of  x  and  y, 
in  which  P  is  any  point  of  y  (x,  y)  =  0,  T  is  at  the  inter- 
section of  x  —  sin  y  with  a  line  through  P  parallel  to  the 
axis  of  y,  and  S  is  at  the  intei'section  of  y  =  sin  x,  with 
a  line  through  P  parallel  to  the  axis  of  x. 

For,  let  the  x  and  y  of  the  trigonometric 
locus  be  written  xf  and  y' ;  then,  if  P  is  a 
point  of  any  locus,  as  y  =  mx,  and  if  S  is 
the  intersection  of  PS  parallel  to  OX  with 
OSHj  the  curve  of  sines  whose  equation 
is  y  =  sin  x',  and  if  T  is  the  intersection 
of  PT  parallel  to  OY  with  OTK,  whose 
equation  is  x  =  sin  y', 


AUXILIARY  CURVES  OF  SINES.  189 


we  have         y  =  AP  =  HS,     .'  .    xf  =  OM=NP', 
and  x  =  OA 


OP'  is  represented  by  sin  x'  =  m  sin  yf.  Hence,  when 
x  and  y  refer  to  P,  x'  and  yf  as  above  constructed  refer  to  P'} 
and  the  rectangle  SPTPf  is  always  a  construction  of  the  relation 
between  a  locus  and  a  trigonometric  locus,  derived  from  it  by  the 
previously  mentioned  substitution. 

486.  Schol.  1.  —  Only  that  part  of  the  locus   <p  (x,  y)  =  0   can  be 

used  in  constructing  <p  (sin  y,  sin  x)  —  0,  which  lies  within  the  square 
whose  sides  are  x=±  1  and  y=  ±  1,  and  every  point  P  within  this 
square  has  a  corresponding  point  P'  within  the  square  x  =  ±  -^TT, 
y  =  ±  gTr,  while  every  point  P  which  is  on  the  side  of  the  square 
x  =  ±l,  y=±l,  has  a  corresponding  point  P'  on  the  side  of  the 
square  x  =  ±  ^TT,  y  =  ±  ^-TT. 

487.  Schol.  2—  When  PT  is  tangent  to  <p  (x,  y)  =  0,  then  P'  T  is 
also  tangent  to  <f>  (sin  y,  sin  x)  —  0  ;   for  as  P  moves  along  its  locus 
parallel  to  the  axis  of  y,  P'  moves  along  its  locus  parallel  to  the  axis 
of  x.     When  PS  is  tangent,  P'S  is  likewise  tangent.     From  this  it 
follows  that  when  the  axis  of  x  is  tangent  to  <p  (x,  y}—0  at  the  origin, 
the  axis  of  y  is  tangent  to  <p  (sin  y,  sin  x)  =  0  at  the  origin. 

A  single  exception  occurs  ;  for  when  PT  is  tangent  to  <f>  (x,  y)  —  0, 
and  PT  is  also  the  line  KE  or  x=±  1,  then  P'  T  is  not  tangent  to 
<p  (sin  y,  sin  x)  =  0,  and  similarly  when  PS  is  tangent,  and  is  the  line 
FH  or  y  =  ±  1,  then  P'/S  is  not  a  tangent. 

Proposition  9. 

488.  Theorem.—  The  equation 

<f>  (sin  y,  sin  x)=0 

represents  a  curve  (or  calico  pattern)  such  that  one  entire 
pattern  which  lies  within  the  square  whose  sides  are  repre- 
sented by  the  equations  x  =  ±  ^~  and  y  =  ±  -'  -  is  in- 
definitely repeated  in  each  direction  in  such  a  manner  as 


190 


TRANSCENDENTAL   CURVES. 


would  be  caused  by  a  kaleidoscope  of  four  mirrors  whose 
cross-section  is  the  square  mentioned. 

For,  it  is  noticeable  that  the  line  PS^  intersects  the  curve 
y  =  sin  x  in  an  infinite  number  of  points  Sl}  S2,  83,  etc.,  on  both 
sides  of  0. 


X 


Also  that  PjPj  intersects  the  curve  »  =  sin  y  in  an  infinite 
number  of  points  Tl}  T2,  T3,  etc.,  above  and  below  0,  and  that 
from  8n  and  Tm  we  can  obtain  Pfnm.  Again,  as  P  is  moved 
along,  <p(x,  y)  =0,  P'n  moves  away  from  the  axes  of  y  at  the 
same  rate  that  P'21  moves  toward  it,  and  vice  versa ;  while  P'12 
moves  away  from  the  axis  of  x  as  P'n  moves  toward  it,  and  vice 
versa,  causing  the  reflected  repetition  spoken  of. 

489.  Schol. — From   Arts.  486    and   487,   if    <p  (»,  y)=°    crosses 
x  =  ±  1   or  y  —  ±:  _?  at  any  angle,  then  <p  (sin  y,  sin  x)  =  0  crosses 
the  sides  of  the  square   x  =  ±  | TT,  y  =  ±  ^-TT  at  right  angles.     That  is, 
the  patterns  of  the  adjacent  squares  join  in  such  a  way  as  to  have  a 
common   tangent   parallel    either    to   the    axis    of  x   or   y.      But   if 
<f>  (x,  y)  =  0   is  tangent  to   x  =  ±  1   or   y—±l,    then  the  correspond- 
ing part  of  v(sin  y,  sin  x)-—0  may  meet  a?  =  db'|*    or    y=±^n 
at  any  angle. 

Proposition  10. 

490.  Theorem.— If    ?  (x,  y)  =  0    be   traced  and    P    be  any 
point  of  it,  and  the  auxiliary  curves  x  =  tan  y   and  y  =  tan  x 
be  also  traced,  then    P'    the  point  of  <p  (tan  y,  tan  x)  =  0    cor- 
responding  to    P   may  be  constructed  as  in  Proposition  8 
by  the  use  of  the  rectangle    SPTP'. 


CURVES  OF  TANGENTS  AS  AUXILIARIES. 


191 


For,  if  P  is  any  point  of  the  locus 
y  —  mx,  and  if  S  is  the  intersection  of 
PAS  parallel  to  OX  with  OSH  whose 
equation  is  y  —  tan  x',  and  if  T  is  the 
intersection  of  PT  parallel  to  OF  with 
OTK  whose  equation  is  x  = 

then  V : 


y 


and 


=  OA=NTy 


ir 


M     F 


.'.     y'=AT=MPf, 
equation  is  <p  (tan  y,  tan  x)  —  0. 


OPf  is  the  curve  whose 


491.  Schol.  1. — All  portions  of  <f>  (x,  y)  =  0  are  used  in  construct- 
ing <p  (tan  y,  tan  x)  =  0,  and  any  finite  point  P  has  a  corresponding 
point  P  within  the  square  x  —  ±  JTTT,  y  =  ±  |  r.     P'  is  ow  the  side  of 
this  square  only  when  P  is  situated  at  infinity. 

492.  Schol.  2—  When  PI7  is  tangent  to  ?  (a?,  y)  -  0,  then  PT  is 
tangent  to  ^  (tan  y,  tan  a?)  —  0,  and  when  PS  is  a  tangent  then  P'/S 
is  also  a  tangent  (cf.  Art.  487). 


Proposition  11. 
493.  TJieorem. — The  equation 

<p  (tan  y,  tan  x)=0 

represents  a  curve  (or  calico  pattern)  such  that  one  entire 
pattern  (which  lies  within  the  square  x=  ±  i  T,  y  =  ±  -*) 
is  repeated  indefinitely  in  each  direction  without  change 
of  form. 

For,  this  may  be  made  to  appear  by  a  demonstration  precisely 
similar  to  that  of  Art.  488. 


494.  Schol. — When    <p  (x,  y)  =  0    has    an    infinite    branch    whose 
approximate  curve  at  infinity  is  parabolic,  then  the  corresponding  part 


192  TRANSCENDENTAL   CURVES. 

of  (f  (tan  y,  tan  x)  =  0  crosses  the  square  #  =  ±:  -TT,  y==±pr  at  a 
corner,  and  at  an  angle  of  4$°  with  the  axes  of  x  and  y — i.  e.,  the 
patterns  of  the  adjacent  squares  join  in  such  a  way  as  to  have  a  common 
tangent  x  =  ±y.  This  appears  from  the  fact  that  the  auxiliary  curves 
of  tangents  are  asymptotic  to  the  lines  x=  ±  ^~,  y  =  ±  ^~. 

J*roposition  12. 

495.  Theorem.— By  the  aid  of  <f>  (x,  y)  —  0,   and  the  auxili- 
ary curves  x  =  sin  y,    «7£<#  y  =  tan  #,   £7^e  locus    y  (sin  y,  tan  x  =  0) 
may  be  constructed,  one  entire  pattern  of  which  lies  within 
the  square    x—±jj-,  y  —  ±j,-,    and  is  repeated  indefinitely 
without  change  of  form   between  the  parallels    y  =  dz  J-  JT, 
#7^0  repetition  being  such  as  would  be  caused  by  a  reflection 
in  two  plane  mirrors  whose  cross-section  is    y  =  ±^^. 

For,  this  may  be  made  to  appear  by  a  mode  of  demonstration 
similar  to  that  of  the  preceding  propositions. 

496.  Schol. — Only  that  part  of  <p(x,y)  =  0  is  used  in  constructing 
<f>  (sin  y,  tan  x)  =  0  which  lies  between  the  parallels  y  =  ±l. 

Proposition'  13. 

497.  Theorem.— By  the  aid  of   <p  (x,  y)  =  0,    and  the  auxili- 
ary curves  x  =  sec  y,   and  y  =  sec  x,  the  locus  <p  (sec  ?/,  sec  x)  =  0 
may  be  constructed,  one  entire  pattern  of  which  lies  within 
the  square    X  =  ±^K,  y=±^n,    and  is  repeated  indefinitely 
in  each  direction  in  such  a  manner  as  would  be  shown  by 
reflecting  from  four  plane   mirrors   whose   cross-section  is 
x  =  ±  ^  ?r    and    y  —  ±  -jir. 

For,  apply  to  this  a  demonstration  similar  to  that  of  the  pre- 
vious propositions. 

498.  Schol. — That  part  of  y  (x,  y)  =  0  which  lies  within  the  square 
x=  ±-%n,  y=±-g  *  cannot  be  used  in  constructing  y  (sec  y,  sec  x)  =  0, 


EXERCISES.  193 


499.  Exercises.  —  Construct  the  following  repeating  curves  : 
(1.)  sin2y  =  4®  tan  x. 

(2.)  sin2x  —  sin  y  —  sin2!/. 

ton*      see 


sin  a      sin  p 


,.  ,  sin2a:  _  sin2?/  = 

sin**      sin»0 
(5.)  cos3x  4-  3a  tan  y  cos  a;  +  tan'y  =  0. 

(6.)  Construct  and  discuss  the  trig,  loci  obtained  from  combinations 
of  trig,  functions  different  from  those  already  discussed. 

E.  G.  <f>  (sec  y,  tan  x)  =  0,     <p  (sec  y,  sin  x)  —  0,  etc.,  etc. 

Trace  the  loci  represented  by  the  following  equations,  and  also  point 
out  the  manner  in  which  the  repetition  occurs.* 

(7.)     sin  x  =  0,    sin  x  =  0.5,    sin  x  =  0.99. 

(8.)     sin2.r  +  sin2y  =  a,  in  which  a  is  successively  0,  0.01,  0.99,  1,  2. 

\ 

(9.)  sin  p  =  r,   in  which  r  is  successively  0,  0.01,  0.99. 

(10.)  sin  i/x  =  a. 

(11.)  Bm(p  +  x)  =  0. 

(12.)  BinG9  +  J)-'=0. 

(13.)  y2  =  (^-a:2)cos^. 

(14.)  sin2#  +  sin2y  +  sinV  —  0.  • 

(15.)  sin  (y  —  sin  ax]  =  0. 

(16.)  sin  (ra  sin  x  sin  y)  =  0. 

(17.)  sin*a?  +  sin^y  =  —x. 

(18.)  sin  x  sin  y  =•  a  sin  -  x  sin  —  y. 

*  Prof.  H.  A.  Newton,  of  Yale  College,  has  invented  repeating  curves  in  great 
number  and  variety.     To  him  these  equations  are  due. 
33 


CHAPTER   XII. 

SPIRALS   AND  POLAR   CURVES. 

500.  A  Spiral  is  the  locus  of  a  point  moving  along  a  line  at  the 
same  time  that  the  line  itself  revolves  about  a  fixed  point,  pro- 
vided that  the  two  motions  have  such  a  relation  that  an  infinite 
number  of  revolutions  of  the  line  must  be  made  to  complete  the 
locus. 

Frequently  the  polar  equation  of  the  spiral  is  in  its  simplest 
form  when  the  fixed  point  is  made  the  pole,  the  distance  of  the 
moving  point  the  radius  vector  and  the  angle  of  revolution  the 
variable  angle. 

By  a  polar  curve  is  usually  signified  one  which  has  for  its  pole 
such  a  point  as  reduces  its  polar  equation  to  some  degree  of  sim- 
plicity. Such  equations  have  been  given  for  the  trisectrix,  lima- 
con,  etc. 

^Proposition  1. 

501.  Theorem.— If  in  the  equation  of  any  algebraic  curve 
(say,     y(x,  y)  =  0),    p  be  written  in  place  of   x,    and    0    in 
place  of  y,    the  equation  thus  obtained    (say,    y(f>,  0)  =  0) 
represents  a  polar  curve  or  spiral  which  may  be   readily 
traced  from  its  relation  to  the  original  curve  (viz.  <p  (x,  y)  =  0). 

For,  the  polar  equation  />  =  ad,  by  putting  x  for  p  and  y  for  6, 

194 


ANALOGOUS  CURVES. 


195 


becomes    x  —  ay.     Let   a  =  j  say.     Draw   the    line 

x  =  \y.     Also   draw  a  circle  with  radius    OA  =  1, 

which  is  the  measuring  circle.  The  arc  6  is  measured 

off  in  linear  units   upon  the  circumference  of   this 
circle. 


A\YX 


When     0'Nl=y  =  0  =  ^n,     then 

When     0'N2  =y  =  0  =  fjr,     then     N2P2=x  =  p  =  -j,ic,  etc.,  etc. 

And  a  similar  construction  may  evidently  be  made  of  any  polar 
curve  from  its  analogous  rectangular  curve. 

502.  SchoL  1. — The  polar  curve  p  —aQ  is  the  spiral  of  Archimedes, 

and  it  is  analogous  to  the  right  line  through  the  origin.  In  it  the 
radius  vector  p  is  evidently  proportional  to  the  arc  0  of  the  measuring 
circle.  Since  each  value  of  0  negative  as  well  as  positive  gives  a  single 
value  of  p,  the  spiral  is  symmetrical  about  the  axis  of  y. 

503.  Schol.  2. — The  circle  />  =  a  is  analogous  to  the  line  x  =  a 
parallel  to  the  axis  of  y,  and  the  fixed  line  0  =  a  is  analogous  to  the 
line  y  =  a  parallel  to  the  axis  of  x. 

504.  Exercises. — Construct  the  spirals, 

(1.)  f-l*P0  +  P  =  0. 


(3.) 


196 


SPIRALS  AND  POLAR   CURVES. 


Proposition  2. 

505.  TJieorem.—  Parabolic  Spirals  represented  by  equations 
of  the  form 


in  which    r    and    t    are  different  positive  integers,  all  have 
the  initial  line  tangent  to  them  at  the  pole. 

For,  let  P2  and  P3  be  upon  the  spiral,  and  let  the  angle 
QPSP2  =  (p.  Then  if  P2P3  is  infinitesimal,  we  may  consider  the 
perpendicular  QP2  =  arc  p^^  —  O^)  as  it  is 


when  02  =  63) 


. 

tan     = 


.  .  (a.) 


3          a 

2    as  in  Art.  423, 


when  P2P3  is  infinitesimal. 
Find  the  value  of 
and  then  make       Pi=P2  =  PS    and     Ol  =  d2  =  03, 
.  ' .     from  (a.)    tan  </>  = 


T 

tan  d>  =—  i 


in  which  (p  is  the  angle  between  the  tangent  and  radius  vector  of 
(ft,  0,).     Letd1  =  0t    then    ft  =  0,    and    tan^  =  0, 
.  * .     the  tangent  to  the  spiral  at  the  pole  has  the  same  direction 
as  the  initial  radius  vector  and  initial  line. 


506.  Schol.  1. — When  r  is  even  and  t  is  odd, 
the  spiral  has  for  every  value  of  0  two  numerically 
equal  values  of  p  with  opposite  signs,  so  that  every 
polar  chord  is  bisected  at  the  pole. 


\ 


PARABOLIC  AND  HYPERBOLIC  SPIRALS. 


197 


507.  Schol.  2. — When  r  is  odd  and  t  is  odd,  the 
curve  is  symmetric  about  the  axis  of  y,  for  — p  and 
—  0  may  be  written  for  p  and  0  without  altering 
the  equation. 


AT; 


508.  Schol.  3. — When  r  is  odd  and  t  is  even,  the 

\/' 
curve  is  symmetric  about  the  axis  of  x,  for  equal       A 

values  of  0  of  opposite  sign  give  the  same  value       y 

\ 

of  p. 


)       X 


509.  Schol.  4=. — When  r  and  t  are  both  even,  the  equation  may 
represent  at  least  two  of  the  previously  mentioned  spirals. 

510.  Schol.  5. — The  spiral  whose  equation  is  /o2  =  ad  is  frequently 
called  the  parabolic  spiral. 

511.  Schol.  6. — If  00  +  0  be  put  for  0  (Art.  49)  in  the  equation  of 
any  parabolic  spiral,  the  curve  is  no  longer  tangent  to  the  initial  line, 
but  is  tangent  instead  to  the  line  0  =  —  00,  for  the  curve  is  rotated 
through  the  angle  —  0Q. 


Proposition  3. 

512.  TJieor  em.  —Hyperbolic  spirals  represented  by  equations 
of  the  form 


in  which  r  and  t  are  positive  integers,  all  approximate 
to  the  initial  line  at  infinity,  and  approach  the  pole  by  an 
infinite  number  of  revolutions  of  the  generating  line. 

ft       ft 

For,  find  the  value  of  —  --  2,  as  in  Art.  427,  and  substitute  it 
ft  -A1 


198 


SPIRALS  AND  POLAR   CURVES. 


in  (a.)  of  Art.  505,  then  make  p±  —  pz  =  /?3  and   Oi  =  02  =  6B. 
.'.     tan^=   -fa (6.). 

Let  Ol  =  0,  then  pl  =  oo,  and  tan  </>  =  0,  .  • .  the  spiral  approxi- 
mates to  the  initial  line  at  infinity.  Again,  if  f>—0,  then  6  =  oo; 
.  * .  the  spiral  approaches  the  pole  as  6  becomes  infinite. 


513.  Schol.  1.— 

When  r  is  even  and 
t  is  odd,  each  polar 
chord  is  bisected  at 
the  pole. 


514.  Schol.  2.— 

When  r  is  odd  and  t 
is  odd,  the  spiral  is 
symmetric  about  the 
axis  of  y. 


X 


515.  Schol.  3. — When  r  is  odd 
and  t  is  even,  the  spiral  is  symmetric 
about  the  axis  of  #. 


516.  Schol.  4. — When  r  is  even  and  t  is  even,  the  equation  may 
represent  two  separate  spirals. 

517.  Schol.  5. — The  spiral   pO  =  a   is   often  called  the  hyperbolic 
spiral,  and  is  asymptotic  at  infinity  to  a  line  parallel  to  the  initial  line 
and  tangent  to  the  circle  p  =  a. 

The    Lituus   pz0  =  a   is  asymptotic  to  the   initial  line  at  infinity, 
having  the  point  of  contraflexure 


THE  EXPONENTIAL  POLYGON  OF  SPIRALS.  199 

518.  Schol.  6.— If  00  +  0  be  substituted  for  0  (Art.  49),  in  the  equa- 
tion of  any  hyperbolic  spiral,  the  curve  no  longer  approximates  to  the 
initial  line  at  infinity,  but  approximates  instead  to  the  line  0  ==  —  0Q. 

519.  The  exponential  polygon  may  be  applied  to  finding  spirals 
which  approximate  in  shape  and  position  to  those  parts  of  polar 
carves  for  which  />  and  6  are  infinite  or  infinitesimal. 

The  same  principles  are  applicable  to  the  determination  of  the  approxi- 
mate equations  which  can  be  derived  from  any  given  equation  in  p  and 
0,  as  were  enunciated  and  proved  in  Chapter  X.  with  respect  to  x  and 
y,  but  the  interpretation  of  the  approximate  equations  so  obtained 
needs  consideration. 

We  shall  state  the  interpretation  of  these  approximate  equations  with- 
out formal  proof,  as  the  interpretation  appears  at  once  from  the  analogies 
already  pointed  out  between  polar  and  rectangular  equations. 

For  convenience  we  shall  consider  that  the  spiral  of  Archimedes  is  one  of  the 
parabolic  spirals. 

520.  Sides. — The  exponential  polygon,  when  applied  to  alge- 
braic equations  in  p  and  0,  has  sides  of  eight  kinds,  viz. : 

1st.  Sides  whose  intercepts  on  the  axis  of  r  and  i  are  both  positive, 
and  which  lie  between  the  origin  and  the  rest  of  the  polygon. 

The  approximate  equation  for  such  a  side  represents  a  parabolic  spiral 
which  approximates  to  the  curve  of  the  original  equation  near  the  pole. 

2d.  Sides  whose  intercepts  on  the  axis  of  r  and  t  are  both  positive, 
and  between  which  and  the  origin  lies  the  rest  of  the  polygon. 

The  approximate  equation  for  such  a  side  represents  a  parabolic  spiral 
which  approximates  to  the  original  curve  for  infinite  values  of  p  and  0. 

3d.  Sides  whose  intercepts  Dispositive  on  the  axis  of  r,  and  negative 
on  the  axis  of  t. 

The  approximate  equation  for  such  a  side  represents  a  hyperbolic 
spiral  which  approximates  to  the  original  curve  for  infinite  values  of  /», 
and  infinitesimal  values  of  0 — i.  e.,  to  the  initial  line  at  infinity. 

4th.  Sides  whose  intercepts  are  negative  on  the  axis  of  r,  and  positive 
on  the  axis  of  t. 

The  approximate  equation  for  such  a  side  represents  an  hyperbolic 
spiral  which  approximates  to  the  original  curve  for  infinitesimal  values 
of  p  and  infinite  values  of  0 — i.  e.,  to  the  pole. 

A  side  whose  intercepts  are  zero  falls  under  the  third  or  fourth  case,  according 
as  all  the  rest  of  the  polygon  is  above  that  side  or  to  the  right  of  it. 


200  SPIRALS  AND  POLAR   CURVES. 

5th.  Sides  that  coincide  with  the  axis  of  r. 

The  approximate  equation  for  such  a  side  gives  the  value  of  p  when 
0  =  0  —  i.  e.,  the  first  intercept  on  the  initial  line. 

6th.  Sides  that  coincide  with  the  axis  of  t. 

The  approximate  equation  for*  such  a  side  gives  the  value  of  0  when 
p  =  0  —  i.  e.,  the  direction  of  the  tangent  at  the  pole. 

7th.  Sides  parallel  to  the  axis  of  r,  and  having  the  rest  of  the  poly- 
gon between  themselves  and  the  origin. 

The  approximate  equation  for  such  a  side  gives  a  constant  value  of  p 
when  0  is  infinite  —  i.  e.,  the  circle  to  which  the  spiral  is  asymptotic. 

8th.  Sides  parallel  to  the  axis  of  t,-  and  having  the  rest  of  the  poly- 
gon between  themselves  and  the  origin. 

The  approximate  equation  for  such  a  side  gives  a  constant  value  of  0 
when  p  is  infinite  —  i.  e.,  the  direction  of  a  radius  vector  which  approxi- 
mates to  the  position  of  an  infinite  branch  of  the  spiral. 

521.  Examples.  —  Find  the  spirals  which  approximate  to 

7r)2  =  a(02-7r2).    (3.)  aY=(p-a)*0\ 


522.  Exercises.  —  (1.)  If  a  line  OT  be  drawn  from  the  pole  0  per- 
pendicular to  the  radius  vector  OP  =  p,  and  intersecting  at  Tthe  tan- 
gent TP,  then  OT=p'  is  the  polar  subtangent  of  P,  and  (Arts.  505, 
512)  P'  =  p  tan  0. 

If  p  =  aO*  ,  show  that  the  locus  of  T  is  p'  =  -(0±j  7:)'+',  in  which  t 
is  positive,  negative,  integral  or  fractional. 

(2.)  Discuss  the  curves  represented  by  the  equation  --  =  sin  -  ; 
they  will  illustrate  the  statements  of  Art.  234. 

It  will  be  found  that  when  T  =  a  there  are  one  or  more  cusps  at  the 
pole,  and  when  r  <  a  the  cusps  are  replaced  by  loops.  E.  G.  When 
n  =  l,  if  r  =  a  the  curve  is  a  cardioid,  but  if  r  =  2a  it  is  a  lima^on. 
If  r  >  a,  the  curve  does  not  pass  through  the  pole. 

Again,  it  will  be  found  that  an  entire  branch  of  the  curve  is  com- 
pleted by  the  revolution  of  p  through  2mt.  If  n  is  commensurable, 
after  the  completion  of  a  certain  number  of  branches,  no  new  branches 
are  described  by  the  further  revolution  of  p.  When  n  is  a  vulgar 
fraction  in  its  lowest  terms,  the  numerator  states  the  number  of  revo- 
lutions of  p  before  the  figure  is  redescribed",  and  the  denominator  the 
number  of  similar  branches  in  the  complete  curve.  If  n  is  incom- 
mensurable, both  these  numbers  are  infinite. 


INITIAL  FINE  OF  25  CENTS 

BOOK 

SS 

OVERDUE. 


S.VSNTH    OAV 


LD  21-100m-12,'43  (8796s) 


•  '. 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


